Target level
Baccalaureate +5
ECTS
120 credits
Duration
2 years
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées), UFR Sciences de l'Homme et de la Société (SHS), Grenoble INP - Ensimag (Informatique, mathématiques appliquées et télécommunications), UGA
Language(s) of instruction
English, French
Presentation
Below is a diagram (in French) of the structure of the master : on the left column, the first year masters (core curriculum), on the center and right columns the second year masters.
Co-accredited training between the Grenoble Alpes University, the Polytechnic Institute of Grenoble, and the University of Savoie Mont-Blanc.
This master courses offers several programs :
- Science industrial applied mathematics (MSIAM) : first year + second year
- Preparation for agregation : second year
- Cybersecurity (CybSec) : second year
- Fondamental mathematics : second year
- Statistics and data science (1) : first year + second year
- Operation recherch combinatorics and optimization (ORCO) : second year
- Mathematical modeling applied analysis (MMAA) (2) : first year + second year
(1) Co-delivered by the Humanities and social sciences teaching department of Grenoble Alpes University
(2) Delivered by the Université de Savoie Mont Blanc
The master proposes two core curricula :
- General mathematics core curriculum in French
- Applied mathematics core curriculum in French and English
Differentiation at first year level : The optional teaching units proposed in semester 7 and semester 8 aim at guiding the students towards the various courses of the second year of the master. The Statistics and data science program is independent of the core curricula. The Mathematical modelling applied analysis program is also independent of the core curricula, but one can enter it at the second year level.
Differentiation of the courses at the second year level (Statistics and data sciences and Mathematical modelling applied analysis excepted) :
- The Science in industrial and applied mathematics, based on the core curriculum Applied mathematics accessible via the core curriculum General mathematics
- Fundamental mathematics, based on the core curriculum General mathematics
- Preparation for agregation, based on the core curriculum General mathematics
- Cybersecurity, accessible via the core curricula Applied mathematics and General mathematics, as well as via the core curriculum Computer science of the Computer science master program
- ORCO, accessible via the core curricula Applied mathematics and General mathematics, as well as via the core curriculum Computer science of the master program Computer science
The objective of this master is to train highly skilled specialists in mathematics and computer science for engineering, teaching, and research in a wide range of fields (pure and applied maths) where the demand from the socio-economic world is strong : security and cryptology, scientific computing, operational research, big data analysis, image synthesis and processing, statistics...
Identifier ROME
IT studies and development
Skills
The basic courses (between 40 and 50 ECTS) are offered in French or English in the first year of the Master.
For research-oriented courses: body of general research-related competencies
- formulate a problem, establish a state of the art, estimate the feasibility, and the impact of a resolution of problem, establish, follow a strategy. Skills are acquired during TER, projects and internships research in M1 and M2 (> 30 ECTS).
Discovery of the socio-economic world offered to all students through introductory modules to the company, project and industrial internships (at least 36 ECTS for career paths), the business forum (presentation of ~ 40 companies, interviews, tables rounds ...) and thematic conferences given by industry.
All students also have access to language courses (English or French as a foreign language depending on their level, 6 ECTS)
International education
- Double degrees, joint degrees, Erasmus Mundus
- Education with formalized international partnerships
- Internationally-oriented programmes
International dimension
- Course CM-BHC in Erasmus Mundus
- CS course, MSIAM are entirely in English, international recruitment
- MF course taught in English according to the public, international recruitment
Organisation
Abroad intership
In France or abroad
Program
Select a program
Preparation for agregation
To view the presentation of the Preparation for agregation program in French click on the following link : Préparation à l'agrégation
Fundamentals mathematics
This is a high-level training in fundamental mathematics research. This course is the gateway to contemporary research in fundamental mathematics, in Grenoble. The master 2nd year honors in Mathematics and mathematical applications pathways of the Fourier institute is part of the Graduate school of mathematics, Information sciences and technologies, Computer science and depends on the University Grenoble Alpes. This course is recommended to students from the 1st year of general mathematics, and candidates to the agregation of mathematics, before they perform their tenure.
The objectives are to have an ntroduction to fundamental mathematics research. Preparation for a PhD thesis.
UE Algebra
9 creditsUE Holomorphic functions
6 creditsUE Probabilities
9 creditsUE Analysis
9 credits
UE Study and research work
6 creditsOptional
UE Effective algebra and cryptographie
6 creditsUE Compléments sur les EDP
6 creditsUE Differential geometry
6 creditsUE Markov process
6 creditsUE Galois theory
6 creditsUE Operations Research (AM)
6 credits
Choice: 1 among 2
UE English S8
3 creditsUE Opening UE (only if C1 level in English reached)
3 credits
Choice: 2 among 3
UE Morse theory in geometry and topology
12 creditsUE Random models on lattices
12 creditsUE Analysis and probability on manifolds
12 credits
Choice: 1 among 2
UE Research internship
27 creditsUE English
Operations Research, combinatorics and optimization (ORCO)
Semester 9 corresponds to the specialization training and semester 10 consists of a practicum in a company or laboratory of 5 to 7 months, which represents 30 European Credit Transfer and Accumulation System credits. The master in Operations research, combinatorics and optimization is one of the possible specializations for the second year of the master of science in Computer science. The courses are taught in English.
The scientific objectives are :
- To train students in the foundations and methods of operational research (mathematical programming, graph theory, complexity, stochastic programming, heuristics, approximation algorithms...)
- To prepare students to use and develop these methods to solve complex industrial applications (supply chain, scheduling, transport, revenue management, etc.) and implement the corresponding software solutions
Students leaving this course equipped to, according to their preferences, move towards the research professions (academic or industrial thesis), enter, as a specialist engineer, major research and development departments in optimization (SNCF, IBM, Air France, Amadeus etc) or enter optimization consulting firms (Eurodécision, Artelys...). They will also be able to enter less specialized companies by highlighting their ability to methodologically analyse operational problems, thus demonstrating that they are potential key elements in the improvement of the company's performance (by linking up with specialized firms or developing in-house methods).
In the longer term, students who are oriented towards the industrial world should be able, with their experience in improving company performance and good "business" knowledge, to naturally access decision-making positions at high levels of responsibility.
The course is labelled "Core AI" by MIAI.
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE English
3 creditsUE Applied probability and statistics
6 creditsUE Systèmes dynamiques
3 creditsUE Instability and Turbulences
3 creditsUE Turbulence
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsChoice: 2 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Statistical analysis and document mining
6 creditsUE Variational methods applied to modelling
6 credits
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE Applied probability and statistics
6 creditsUE English
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsUE GS_MSTIC_Scientific approach
6 creditsChoice: 1 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Variational methods applied to modelling
6 creditsUE Statistical analysis and document mining
6 credits
UE Algebra
9 creditsUE Holomorphic functions
6 creditsUE Probabilities
9 creditsUE Analysis
9 credits
UE Study and research work
6 creditsChoice: 3 among 6
UE Effective algebra and cryptographie
6 creditsUE Compléments sur les EDP
6 creditsUE Differential geometry
6 creditsUE Markov process
6 creditsUE Galois theory
6 creditsUE Operations Research (AM)
6 credits
Choice: 1 among 2
UE English S8
3 creditsUE Opening UE (only if C1 level in English reached)
3 credits
UE Advanced models and methods in operations research
6 creditsUE Combinatorial optimization and graph theory
6 creditsUE Optimization under uncertainty
6 creditsUE Constraint Programming, applications in scheduling
3 creditsUE Graphs and discrete structures
3 creditsUE Advanced heuristic and approximation algorithms
3 creditsUE Advanced mathematical programming methods
3 creditsUE Academic and industrial challenges
3 creditsUE Transport Logistics and Operations Research
6 creditsUE Advanced parallel system
6 creditsUE Multi-agent systems
3 creditsUE Fundamentals of Data Processing and Distributed Knowledge
6 creditsUE Scientific Methodology, Regulatory and ethical data usage
6 creditsUE Large scale Data Management and Distributed Systems
6 creditsUE Cryptographic engineering, protocols and security models, data privacy, coding and applications
6 creditsUE From Basic Machine Learning models to Advanced Kernel Learning
6 creditsUE Mathematical Foundations of Machine Learning
3 creditsUE Learning, Probabilities and Causality
6 creditsUE Statistical learning: from parametric to nonparametric models
6 creditsUE Mathematical optimization
6 creditsUE Safety Critical Systems: from design to verification
6 creditsUE Natural Language Processing & Information Retrieval
6 creditsUE Information Security
6 creditsUE Human Computer Interaction
6 creditsUE Next Generation Software Development
6 credits
UE Practicum
30 credits
UE Advanced models and methods in operations research
6 creditsUE Combinatorial optimization and graph theory
6 creditsUE Optimization under uncertainty
6 creditsUE GS_MSTIC_Research ethics
6 creditsUE Constraint Programming, applications in scheduling
3 creditsUE Graphs and discrete structures
3 creditsUE Advanced heuristic and approximation algorithms
3 creditsUE Advanced mathematical programming methods
3 creditsUE Academic and industrial challenges
3 creditsUE Transport Logistics and Operations Research
6 credits
UE Practicum
30 credits
Cybersecurity
The global economic impact of losses due to cybercrime amounts to hundreds of billions of euros per year ($445 billion according to the McAfee/CSIS study of 2014) with a strong increase in attacks, especially for identity theft and digital data theft, as well as malicious attacks.
Protection against these vulnerabilities includes :
- Robustness to cyber attacks of sensitive infrastructure (e.g. stuxnet)
- Robustness of security components against software vulnerabilities and data leaks (e.g. heartbleed)
- Protection of privacy and security of cloud infrastructure
- Robust design and evaluation of safety components
- Fault detection in protocols or software and hardware components
The topics covered in the training cover the complementary areas of Cybersecurity, including cryptology, forensics, and privacy, in particular for embedded systems and distributed architecture.
The objective of this program is to train cybersecurity experts (including data privacy aspects) with a bac + 5 degree, able to evolve immediately in an industrial environment and who can also pursue a thesis.
The course is labelled "Core AI" by MIAI.
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE English
3 creditsUE Applied probability and statistics
6 creditsUE Systèmes dynamiques
3 creditsUE Instability and Turbulences
3 creditsUE Turbulence
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsChoice: 2 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Statistical analysis and document mining
6 creditsUE Variational methods applied to modelling
6 credits
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE Applied probability and statistics
6 creditsUE English
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsUE GS_MSTIC_Scientific approach
6 creditsChoice: 1 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Variational methods applied to modelling
6 creditsUE Statistical analysis and document mining
6 credits
UE Algebra
9 creditsUE Holomorphic functions
6 creditsUE Probabilities
9 creditsUE Analysis
9 credits
UE Study and research work
6 creditsChoice: 3 among 6
UE Effective algebra and cryptographie
6 creditsUE Compléments sur les EDP
6 creditsUE Differential geometry
6 creditsUE Markov process
6 creditsUE Galois theory
6 creditsUE Operations Research (AM)
6 credits
Choice: 1 among 2
UE English S8
3 creditsUE Opening UE (only if C1 level in English reached)
3 credits
UE Software security, secure programming and computer forensics
3 creditsUE Security architecture
6 creditsUE Cryptographic engineering, protocols and security models, data privacy, coding and applications
6 creditsUE Threat and risk analysis, IT security audit and norms
3 creditsUE Physical Security : Embedded, Smart Card, Quantum & Biometrics
6 creditsChoice: 1 to 2 among 2
UE Advanced cryptology
6 creditsUE Advanced security
6 credits
UE Software security, secure programming and computer forensics
3 creditsUE Cryptographic engineering, protocols and security models, data privacy, coding and applications
6 creditsUE Threat and risk analysis, IT security audit and norms
3 creditsUE Physical Security : Embedded, Smart Card, Quantum & Biometrics
6 creditsUE GS_MSTIC_Research ethics
6 creditsChoice: 1 among 2
UE Advanced cryptology
6 creditsUE Advanced security
6 credits
Statistics and data sciences (SSD)
To view the presentation of the Statistics and data sciences (SSD) program in French click on the following link : Parcours Statistiques et sciences de sonnées (SSD)
Science in industrial and applied mathematics (MSIAM)
Currently, applied mathematics is an area that provides many job opportunities, in industry and in the academic world. There is a great demand for mathematical engineers on topics such as scientific computation, big data analysis, imaging and computer graphics, with applications in many fields such as physics, medicine, biology, engineering, finance, environmental sciences.
The master of Science in industrial and applied mathematics (MSIAM) offers a large spectrum of courses, covering areas where the research in applied math in Grenoble is at the best level. The graduates are trained to become experts and leaders in scientific and technological projects where mathematical modeling and computing issues are central, in industry or research. A large and distinguished graduate Faculty participate in the program, bringing their expertise in a wide range of areas of mathematics including applied analysis, numerical analysis and scientific computing, probability theory and statistics, computational graphics, image analysis and processing, and applied geometry.
The academic program is a two-year master program (120 ECTS), fully taught in English. It combines three semesters of courses and laboratory work (90 ECTS) with a six-month individual research project (30 ECTS). The first year is composed of a common core which provides theoretical and practical grounds in probability and statistics, PDE and modelling, images and geometry as well as computer sciences, optimisation and cryptology.
In the second year, the third semester is divided in 2 tracks :
- Modeling, Scientific Computing and Image analysis (MSCI)
- Data Science (DS)
The semester 10 is devoted to the master thesis project.
The course is labelled "Core AI" by MIAI.
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE English
3 creditsUE Applied probability and statistics
6 creditsUE Systèmes dynamiques
3 creditsUE Instability and Turbulences
3 creditsUE Turbulence
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsChoice: 2 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Statistical analysis and document mining
6 creditsUE Variational methods applied to modelling
6 credits
UE Object-oriented and software design
3 creditsUE Partial differential equations and numerical methods
6 creditsUE Signal and image processing
6 creditsUE Geometric modelling
6 creditsUE Applied probability and statistics
6 creditsUE English
3 credits
UE Computing science for big data and HPC
6 creditsHPC
Introduction to database
3 credits
UE Project
3 creditsUE Internship
3 creditsUE Numerical optimisation
6 creditsUE GS_MSTIC_Scientific approach
6 creditsChoice: 1 among 6
UE Operations Research (AM)
6 creditsUE Introduction to cryptology (AM)
6 creditsUE 3D Graphics (AM)
6 creditsUE Turbulences
6 creditsUE Variational methods applied to modelling
6 creditsUE Statistical analysis and document mining
6 credits
UE Differential Calculus, Wavelets and Applications
6 creditsUE An Introduction to Shape and Topology Optimization
3 creditsUE Efficient methods in optimization
3 creditsUE Computational biology
3 creditsUE Fluid Mechanics and Granular Materials
6 creditsUE GPU Computing
6 creditsUE Software development tools and methods
3 creditsUE Geophysical imaging
3 creditsUE Handling uncertainties in (large-scale) numerical models
6 creditsUE Modeling seminar and projects
6 creditsUE Quantum Information & Dynamics
6 creditsUE Optimal transport: theory, applications and related numerical methods
6 creditsUE Statistical learning: from parametric to nonparametric models
6 creditsUE Temporal, spatial and extreme event analysis
6 credits
UE Research projects
30 credits
UE Advanced Machine Learning: Applications to Vision, Audio and Text
6 creditsUE An Introduction to Shape and Topology Optimization
3 creditsUE Computational biology
3 creditsUE Data Science Seminars and Challenge
6 creditsUE Differential Calculus, Wavelets and Applications
6 creditsUE Efficient methods in optimization
3 creditsUE From Basic Machine Learning models to Advanced Kernel Learning
6 creditsUE Handling uncertainties in (large-scale) numerical models
6 creditsUE GPU Computing
6 creditsUE Learning, Probabilities and Causality
6 creditsUE Mathematical Foundations of Machine Learning
3 creditsUE Modeling seminar and projects
6 creditsUE Optimal transport: theory, applications and related numerical methods
6 creditsUE Natural Language Processing & Information Retrieval
6 creditsUE Statistical learning: from parametric to nonparametric models
6 creditsUE Software development tools and methods
3 creditsUE Temporal, spatial and extreme event analysis
6 credits
UE Research projects
30 credits
UE GS_MSTIC_Research ethics
6 creditsUE Software development tools and methods
3 creditsUE Modeling seminar and projects
6 creditsUE Geophysical imaging
3 creditsUE An Introduction to Shape and Topology Optimization
3 creditsUE Refresh courses
0 creditsUE GPU Computing
6 creditsUE Differential Calculus, Wavelets and Applications
6 creditsUE Optimal transport: theory, applications and related numerical methods
6 creditsUE Fluid Mechanics and Granular Materials
6 creditsUE Handling uncertainties in (large-scale) numerical models
6 creditsUE Temporal, spatial and extreme event analysis
6 creditsUE Advanced Machine Learning: Applications to Vision, Audio and Text
6 creditsUE Natural Language Processing & Information Retrieval
6 creditsUE From Basic Machine Learning models to Advanced Kernel Learning
6 creditsUE Mathematical Foundations of Machine Learning
3 creditsUE Statistical learning: from parametric to nonparametric models
6 creditsUE Learning, Probabilities and Causality
6 creditsUE Efficient methods in optimization
3 creditsUE Data Science Seminars and Challenge
6 creditsUE Computational biology
3 creditsUE Quantum Information & Dynamics
6 creditsUE Numerical Mechanics
6 credits
UE Research projects
30 credits
UE Algebra
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Holomorphic functions
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
- Holomorphic and analytical functions, in particular the equivalence between the two notions, exponential function and logarithm, principle of analytic continuation, principle of isolated zeros, Cauchy formula for the disc
- Elemental properties of holomorphic functions (Cauchy inequalities, sequences and series of holomorphic functions, property of the mean, and principle of the maximum)
- Cauchy theory (existence of primitives, Cauchy theorems)
- Meromorphic functions (classification of isolated singularities, meromorphic functions, residue theorem, Laurent series)
- Riemann conformal representation theorem
UE Probabilities
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Analysis
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Study and research work
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This Teaching Unit proposes a discovery of research in mathematics through the study of a subject describing a result or a mathematical theory, with which the student will have to familiarize themselves in order to appropriate them and to be able to account for them in a written report and an oral presentation.
In practice, a list of subjects is proposed during the first semester. Each student selects four subjects from this list, ranked from 1 to 4, and then the person responsible for the training assigns to each student a topic as far as possible among these four. As soon as the assignments are known, each student will contact the author of their subject, who will supervise them for this work throughout the second semester. Once the supervisor has presented the subject and the details of the expected work, the student meets them regularly to report on the progress of his/her work and progress.
The Supervised Research Work leads to the writing of a written report using LaTeX software, which must include an abstract and a bibliography, and an oral defence of 20 to 30 minutes, often followed by questions, before a jury which includes the supervisor. The report and the defence jointly contribute to the evaluation of the work carried out.
UE Effective algebra and cryptographie
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Compléments sur les EDP
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Compléments du premier semestre : principe du maximum faible pour les ÉDP elliptiques du second ordre, inégalité de Harnack
Compléments sur les espaces de Sobolev : injections de Sobolev, opérateurs d'extension, théorie des traces. Introduction aux distributions, distributions tempérées
Opérateurs maximaux monotones, théorème de Hille-Yosida
Équation de la chaleur dans Ω × ]0,+∞[, où Ω ⊂ ℝn est un domaine régulier : existence et unicité des solutions avec conditions au bord de Dirichlet et de Neumann ; principe du maximum pour les solutions de l'équation de la chaleur
Équation des ondes dans Ω × ]0,+∞[ : existence et unicité des solutions, propagation à vitesse finie
Équation de la chaleur semi-linéaire
UE Differential geometry
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to curves and surfaces.
Curves: Frénet references, covariant derivatives, reference fields, connection forms, structural equations.
Surfaces: surfaces of R^3, plane tangents, differential forms, differentiable applications between surfaces.
Abstract Variety: Whitney's Theorem
Curvature: Normal curvature, Gauss curvature, Geodesics. Case of surfaces of revolution.
Geometry of the surfaces of R^3: Egregium theorem, Gauss Bonnet's theorem
Possibility of complement: Transition to sub-varieties of Rn and abstract varieties
Possibility of complement: Introduction to dynamic systems. Vector fields and dynamic systems on varieties
UE Markov process
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Galois theory
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE English S8
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Opening UE (only if C1 level in English reached)
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Morse theory in geometry and topology
Level
Baccalaureate +5
ECTS
12 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Random models on lattices
Level
Baccalaureate +5
ECTS
12 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Analysis and probability on manifolds
Level
Baccalaureate +5
ECTS
12 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Topology of random hypersurfaces
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Probabilistic and geometric techniques in high dimension
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Research internship
Level
Baccalaureate +5
ECTS
27 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Final year research project.
UE English
Level
Baccalaureate +5
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
(if level B2 not obtained)
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course provides an introduction to the basic concepts of object-oriented programming in C++: classes and encapsulation, operator overloading, generic classes (templates), STL (Standard Template Library), inheritance and derived classes, polymorphism and virtual functions.
Lab sessions illustrate these concepts, and applications for applied mathematics.
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Connaitre les notions élémentaires et indispensables de probabilités. Maitriser les calculs de lois, d'espérance; connaitre les fonctions de simulation de logiciels spécialisés; savoir utiliser ces fonctions pour illustrer les résultats
de probabilité. Chaque notion sera abordée en cours et illustrée numériquement en séance
avec les étudiants.
Connaitre les principaux estimateurs, leurs propriétés.
Être capable de programmer les différents estimateurs dans des
situations simples; savoir illustrer par simulation les propriétés des estimateurs (biais, erreur quadratique).
Chaque notion sera abordée en cours et illustrée numériquement en séance avec les étudiants.
Contenu : Lois fondamentales discrètes et continues; Variables indépendantes, lois conditionnelles.
Espérance, espérance conditionnelle. Convergence de variables aléatoires. Théorème
centrale limite, loi des grands nombres. Vecteur gaussien, théorème Cochran. Notion
de chaine de Markov illustrée dans des TP.
Estimateur, estimation. Biais, erreur quadratique, convergence. Intervalle de
confiance. Estimateur des moindres carrés. Estimateur des moments. Estimateur du maximum
de vraisemblance. Equations estimantes.
UE Systèmes dynamiques
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce cours présente des manières géométriques de traiter et résoudre des problèmes décrits par des équations différentielles.
- chapitre I : Introduction : généralités sur les systèmes dynamiques
- chapitre II : Systèmes unidimensionnels : Les points fixes, linéarisation et stabilité, Exemple : le modèle logistique, Existence et unicité des solutions d'équations différentielles ordinaires
- chapitre III : Bifurcations : Bifurcation selle-nœud, Bifurcation transcritique, Bifurcation transcritique imparfaite, Bifurcation fourche, Bifurcation fourche supercritique, Bifurcation fourche sous-critique, Bifurcation fourche supercritique imparfaite
- chapitre IV : Champ de vecteur sur un cercle : Oscillateur uniforme, Oscillateur non-uniforme
- chapitre V : Flots bidimensionnels et applications : Existence et unicité des solutions et conséquences topologiques, Systèmes linéaires, Systèmes non-linéaires : linéarisation proche des points fixes, Cycles limites, Le théorème de Poincaré-Bendixson, Systèmes Liénard, Systèmes gradients, Fonctions de Liapunov
- chapitre VI : Bifurcations bidimensionnelles : Bifurcations selle-nœud, transcritique et fourche, Bifurcation de Hopf, Bifurcations globales de cycles
UE Instability and Turbulences
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
This course presents basic notions on instabilities and turbulence. We try to be as progressive as possible and to base our presentation on analyses of real experiments and real flows. We review few mathematical methods to analyze nonlinear systems in terms of instabilities. The students have to use their new knowledge to run and analyze numerical simulations of very simple systems. We then study some of the most important physical mechanisms for fluid instabilities and the corresponding criteria. We quickly present a zoology of common fluid instabilities and discuss the mechanisms and the possible technical implications. We give a broad introduction on turbulence and describe few fundamental methods and results, in particular the Richardson cascade, the Reynolds decomposition and the Kolmogorov spectra.
Teaching program:
1. General introduction
- Instabilities and turbulence, interest?
- Reynolds experiment and Reynolds number
- Incompressible Navier-Stokes equations: diffusion and advection
- An example: the wake of a cylinder
2. Instabilities and transition to turbulence
- Systems with few degrees of freedom
- Fluid instability mechanisms and conditions
- Other flows examples
3. Effects of variable density
- Boussinesq approximation
- Unstable stratification, Rayleigh-Taylor instability
- Rayleigh-Benard instability (Ra, Nu)
- Stable stratification, Kelvin-Helmoltz instability and Richardson number
4. Turbulence
- Introduction, Richardson cascade
- Average and Reynolds decomposition
- Experimental and numerical methods to study turbulence
- Statistical descriptions
For this course, the students have to write in LaTeX a report on their practical work. Thus, we spent some time for a first gentle introduction of this tool widely used in scientific academia.
UE Turbulence
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce module est une introduction à la turbulence phénoménologique et statistique. On s’intéresse aux définitions et propriétés de la turbulence en terme de processus physiques et leur description dans des familles types d’écoulements cisaillés que l’on peut retrouver dans la nature et en ingénierie.
jet turbulent
jet turbulent
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course provides an introduction to the basic concepts of object-oriented programming in C++: classes and encapsulation, operator overloading, generic classes (templates), STL (Standard Template Library), inheritance and derived classes, polymorphism and virtual functions.
Lab sessions illustrate these concepts, and applications for applied mathematics.
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Connaitre les notions élémentaires et indispensables de probabilités. Maitriser les calculs de lois, d'espérance; connaitre les fonctions de simulation de logiciels spécialisés; savoir utiliser ces fonctions pour illustrer les résultats
de probabilité. Chaque notion sera abordée en cours et illustrée numériquement en séance
avec les étudiants.
Connaitre les principaux estimateurs, leurs propriétés.
Être capable de programmer les différents estimateurs dans des
situations simples; savoir illustrer par simulation les propriétés des estimateurs (biais, erreur quadratique).
Chaque notion sera abordée en cours et illustrée numériquement en séance avec les étudiants.
Contenu : Lois fondamentales discrètes et continues; Variables indépendantes, lois conditionnelles.
Espérance, espérance conditionnelle. Convergence de variables aléatoires. Théorème
centrale limite, loi des grands nombres. Vecteur gaussien, théorème Cochran. Notion
de chaine de Markov illustrée dans des TP.
Estimateur, estimation. Biais, erreur quadratique, convergence. Intervalle de
confiance. Estimateur des moindres carrés. Estimateur des moments. Estimateur du maximum
de vraisemblance. Equations estimantes.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE GS_MSTIC_Scientific approach
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Algebra
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Holomorphic functions
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
- Holomorphic and analytical functions, in particular the equivalence between the two notions, exponential function and logarithm, principle of analytic continuation, principle of isolated zeros, Cauchy formula for the disc
- Elemental properties of holomorphic functions (Cauchy inequalities, sequences and series of holomorphic functions, property of the mean, and principle of the maximum)
- Cauchy theory (existence of primitives, Cauchy theorems)
- Meromorphic functions (classification of isolated singularities, meromorphic functions, residue theorem, Laurent series)
- Riemann conformal representation theorem
UE Probabilities
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Analysis
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Study and research work
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This Teaching Unit proposes a discovery of research in mathematics through the study of a subject describing a result or a mathematical theory, with which the student will have to familiarize themselves in order to appropriate them and to be able to account for them in a written report and an oral presentation.
In practice, a list of subjects is proposed during the first semester. Each student selects four subjects from this list, ranked from 1 to 4, and then the person responsible for the training assigns to each student a topic as far as possible among these four. As soon as the assignments are known, each student will contact the author of their subject, who will supervise them for this work throughout the second semester. Once the supervisor has presented the subject and the details of the expected work, the student meets them regularly to report on the progress of his/her work and progress.
The Supervised Research Work leads to the writing of a written report using LaTeX software, which must include an abstract and a bibliography, and an oral defence of 20 to 30 minutes, often followed by questions, before a jury which includes the supervisor. The report and the defence jointly contribute to the evaluation of the work carried out.
UE Effective algebra and cryptographie
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Compléments sur les EDP
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Compléments du premier semestre : principe du maximum faible pour les ÉDP elliptiques du second ordre, inégalité de Harnack
Compléments sur les espaces de Sobolev : injections de Sobolev, opérateurs d'extension, théorie des traces. Introduction aux distributions, distributions tempérées
Opérateurs maximaux monotones, théorème de Hille-Yosida
Équation de la chaleur dans Ω × ]0,+∞[, où Ω ⊂ ℝn est un domaine régulier : existence et unicité des solutions avec conditions au bord de Dirichlet et de Neumann ; principe du maximum pour les solutions de l'équation de la chaleur
Équation des ondes dans Ω × ]0,+∞[ : existence et unicité des solutions, propagation à vitesse finie
Équation de la chaleur semi-linéaire
UE Differential geometry
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to curves and surfaces.
Curves: Frénet references, covariant derivatives, reference fields, connection forms, structural equations.
Surfaces: surfaces of R^3, plane tangents, differential forms, differentiable applications between surfaces.
Abstract Variety: Whitney's Theorem
Curvature: Normal curvature, Gauss curvature, Geodesics. Case of surfaces of revolution.
Geometry of the surfaces of R^3: Egregium theorem, Gauss Bonnet's theorem
Possibility of complement: Transition to sub-varieties of Rn and abstract varieties
Possibility of complement: Introduction to dynamic systems. Vector fields and dynamic systems on varieties
UE Markov process
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Galois theory
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE English S8
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Opening UE (only if C1 level in English reached)
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Advanced models and methods in operations research
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course presents advanced methods and technics for Operations Research.
Reminder :
Linear Programming, Dynamic Programming, MIP modelling and BB
Complexity (P, NP, Co-NP)
Advanced MIP :
formulation, cuts, bounds
applications
lagragian relaxation
column generation
Benders decomposition
Solvers
Constraint Programming
Heuristics
local search
approximation algorithms
UE Combinatorial optimization and graph theory
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to provide a broad knowledge of fundamental problems in Combinatorial Optimization to show their algorithmic solutions and to derive min-max results on them. In order to achieve this goal a new object called a polyhedron is introduced. This polyhedral approach helps to shed new light on some classic results of Combinatorial Optimization.
Syllabus: Study of polyhedra associated to problems of Combinatorial Optimization ; Theory of blocking polyhedra ; Connectivity: shortest paths, spanning trees and spanning arborescences of minimum weight ; Flows: Edmonds-Karp Algorithm, Goldberg-Tarjan Algorithm, minimum cost flows ; Matchings: Hungarian method, Edmonds' Algorithm, Chinese postman problem; Matroids: greedy algorithm, intersection of two matroids ; Graph coloring ; Applications coming from various areas of Operations Research.
UE Optimization under uncertainty
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The objective of this course is to present different techniques to handle uncertainty in decision problems. These techniques will be illustrated on several applications e.g. inventory control, scheduling, energy, machine learning.
Syllabus : Introduction to uncertainty in optimization problems; Reminders (probability, dynamic programming, ...); Markov chains; Markov decision processes; Stochastic programming; Robust optimization
UE Constraint Programming, applications in scheduling
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Graphs and discrete structures
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to learn how to use the structure of graphs and other discrete objects to obtain general results on them, and in particular efficient algorithms solving important problems.
We will cover the following topics:
- Structural Graph Theory: we will study the structure of important graph classes with nice algorithmic properties (planar graphs, interval graphs, comparability graphs, ...) and show several concrete problems that can be solved thanks to their structural properties.
- Graph Drawing: with the rise of Big Data, representing huge data sets is a fundamental challenge. Efficient ways to represent large graphs will be presented.
- Codes: we will see various codes (dominating, locating, identifying, ...) in graphs and their applications.
- Extremal combinatorics: the typical question in this field is "what global condition do we need to impose in some graph in order to make sure that some nice structure appears locally?" We will introduce a powerful tool called "the probabilistic method", an show how it can be applied to solve problems in this important area of research.
UE Advanced heuristic and approximation algorithms
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Advanced mathematical programming methods
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Academic and industrial challenges
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course offers the possibility for the students to gain some experience by facing open/difficult combinatorial problems.
The goal is to model and solve a combinatorial problem with direct industrial applications. We expect the students to take a variety of approaches (local search, compact/extended linear programming formulations, constraint programming, ...) and establish useful results (lower bounds, cuts, complexity,...).
The experimental results will be compared to the litterature (a known academic open benchmark will be available in this case) or will be validated by the industrial partner.
UE Transport Logistics and Operations Research
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Advanced parallel system
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Today, parallel computing is omnipresent across a large spectrum of computing platforms, from processor cores or GPUs up to Supercomputers and Cloud platforms. The largest Supercomputers gather millions of processing units and are heading towards Exascale (a quintillion or 10^18 flops - http://top500.org). If parallel processing was initially targeting scientific computing, it is now part of many domains like Big Data analytics and Deep Learning. But making an efficient use of these parallel resources requires a deep understanding of architectures, systems, parallel programming models, and parallel algorithms.
Overview:
The class is organized around weekly lectures, discussions and help time. The presented materials that will be available each week on the class web page. To get practical experience and good understanding of the main concepts the students are expected to develop short programs and experiments. Students will also have to prepare a presentation or a written report on research articles. Students will have access to Grid'5000 parallel machines and the SimGrid simulator for experiments.
This class is organized around 2 main blocks:
Overview of parallel systems:
-
- Introduction to parallelism from GPU to supercomputers.
- Hardware and system considerations for parallel processing (multi-core architectures, process and thread handling, cache efficiency, remote data access, atomic instructions)
- Parallel programming: message passing, one-sided communications, task-based programming, work stealing based runtimes (MPI, Cilk, TBB, OpenMP).
- Modeling of parallel programs and platforms. Locality, granularity, memory space, communications.
- Parallel algorithms, collective communications, topology aware algorithms.
- Scheduling: list algorithms; partitioning techniques. Application to resource management (PBS, LSF, SGE, OAR).
- Large scale scientific computing: parallelization of Lagrangian and Eulerian solvers, parallel data analytics and scientific visualization.
- AI and HPC: how parallelism is used at different levels to accelerate machine learning and deep learning using supercomputers.
Functional parallel programming:
We propose to study a clean and modern approach to the design of parallel algorithms: functional programming. Functional languages are known to provide a different and cleaner programming experience by allowing a shift of focus from "how to do" on "what to do".
If you take for example a simple dot product between two vectors. In c language you might end up with:
unsigned int n = length(v1); double s = 0.0; for (unsigned int i = 0 ; i < n ; i++) { s += v1[i] * v2[i]; }
In python however you could write:
return sum(e1*e2 for e1, e2 in zip(v1, v2))
You can easily notice that the c language code displayed here is highly sequential with a data-flow dependence on the i variable. It intrinsically contains an ordering of operations because it tells you how to do things to obtain the final sum. On the other end the python code exhibits no dependencies at all. It does not tell you how to compute the sum but just what to compute: the sum of all products.
In this course we will study how to express parallel operations in a safe and performant way. The main point is to study parallel iterators and their uses but we will also consider classical parallel programming schemes like divide and conquer. We will both study the theoretical complexity of different parallel algorithms and practice programming and performance analysis on real machines.
All applications will be developed in the RUST programming language around the Rayon parallel programming library.
No previous knowledge of the rust language is required as we will introduce it gradually during the course. You need however to be proficient in at least one low level language (typically C or C++)
UE Multi-agent systems
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Multi-agent systems (MAS) is a very active field of AI research, with multiple industrial and societal applications. The 2 main fields of application concern Distributed Problem Solving (DPS) and Agent-Based Modelling and Simulation (ABMS). The goal of this course is to understand the concepts of agents, multi-agent systems, models and simulations, and to learn how to design such models.
This course introduces the field of MAS, various theoretical aspects (agent architectures, reasoning, interactions, game theory, social choice, etc), as well as practical applications from recent research. The focus is mostly on agent-based social simulation, and how to integrate psychological aspects in agents (so-called “human factors”: emotions, biases…) to make them more human-like and realistic. Applications discussed include epidemics modelling, computational economy, crisis management, urban planning, serious games, etc. The practical part of the course comprises several tutorials with various agent-based modelling platforms (in particular GAMA and Netlogo), scientific papers discussions, and analysis and/or extension of existing models.
UE Fundamentals of Data Processing and Distributed Knowledge
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Modern computing increasingly takes advantage of large amounts of distributed data and knowledge. This is grounded on theoretical principles borrowing to several fields of computer science such as programming languages, data bases, structured documentation, logic and artificial intelligence. The goal of this course is to present some of them, the problems that they solve and those that they uncover. The course considers two perspectives on data and knowledge: interpretation (what they mean), analysis (what they reveal) and processing (how can they be traversed efficiently and transformed safely).
The course offers a semantic perspective on distributed knowledge. Distributed knowledge may come from data sources using different ontologies on the semantic web, autonomous software agents learning knowledge or social robots interacting with different interlocutors. The course adopts a synthetic view on these. It first presents principles of the semantics of knowledge representation (RDF, OWL). Ontology alignments are then introduced to reduce the heterogeneity between distributed knowledge and their exploitation for answering federated queries is presented. A practical way for cooperating agents to evolve their knowledge is cultural knowledge evolution that is then illustrated. Finally, the course defines dynamic epistemic logics as a way to model the communication of knowledge and beliefs.
The course also introduces a perspective on programming language foundations, algorithms and tools for processing structured information, and in particular tree-shaped data. It consists in an introduction to relevant theoretical tools with an application to NoSQL (not only SQL) and XML technologies in particular. Theories and algorithmic toolboxes such as fundamentals of tree automata and tree logics are introduced, with applications to practical problems found for extracting information. Applications include efficient query evaluation, memory-efficient validation of document streams, robust type-safe processing of documents, static analysis of expressive queries, and static type-checking of programs manipulating structured information. The course also aims at presenting challenges, important results, and open issues in the area.
UE Scientific Methodology, Regulatory and ethical data usage
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course aims to provide the fundamental basis for a sound scientific methodology of experimental evaluation in computer science. This lecture emphasizes on methodological aspects of measurement and on the statistics needed to analyze computer systems, human-computer interaction systems, and machine learning systems. We first sensibilize the audience to reproducibility issues related to empirical research in computer science as well as to ethical and scientific integrity aspects. Then we present tools that help address the aforementioned issues and we give the audience the basis of probabilities and statistics required to develop sound experiment designs. The content of the lecture is therefore both theoretical and practical, illustrated by a lot of case studies and practical sessions. The goal is not to provide analysis recipes or techniques that researchers can blindly apply but to make students develop critical thinking and understand some simple (and possibly not-so-simple) tools so that they can both readily use and explore later on.
UE Large scale Data Management and Distributed Systems
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is divided in two complementary parts: distributed systems and data management.
Part 1: Distributed systems
Summary
Distributed systems are omnipresent. They are formed by a set of computing devices, interconnected by a network, that collaborate to perform a task. Distributed systems execute on a wide range of infrastructures: from Cloud datacenters to wireless sensor networks.
The goal of this course is to study the main algorithms used at the core of Distributed systems. These algorithms must be efficient and robust. An algorithm is efficient when it sustains a high level of performance. Performance can be measured using various metrics such as throughput, latency, response time. An algorithm is robust when it is able to operate despite the occurrence of various types of (network and/or machine) and attacks.
Content
During the course, we will cover several topics that are listed below:
- Event-driven formalisms for distributed algorithms
- Basic abstractions: processes, links
- Failure detector algorithms
- Leader election algorithms
- Broadcasting algorithms
- Distributed shared memory algorithms
- Consensus algorithms
- Epidemic algorithms
- Performance models for distributed systems
Part 2: Data management
Summary
The ability to process large amounts of data is key to both industry and research today. As computing systems are getting larger, they generate more data that need to be analyzed to extract knowledge.
Data management infrastructures are growing fast, leading to the creation of large data centers and federations of data centers. Suitable software infrastructures should be used to store and process data in this context. Big Data software systems are built to take advantage of a large set of distributed resources to efficiently process massive amounts of data while being able to cope with failures that are frequent at such a scale.
In addition to the amount of data to be processed, the other main challenge that such Big Data systems need to deal with is time. For some use cases, the earlier the results of a data analysis is obtained, the more valuable the result is. Some Big Data systems especially target stream processing where data are processed in real time.
Through lectures and practical sessions, this course provides an overview of the software systems that are used to store and process data at large scale. The following topics will be covered:
- Map-Reduce programming model
- In-memory data processing
- Stream processing (data movement and processing)
- Large scale distributed data storage (distributed file systems, NoSQL databases)
Throughout the lectures, the challenges associated with performance and fault tolerance will also be discussed.
UE Cryptographic engineering, protocols and security models, data privacy, coding and applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course present the main cryptographic primitives and security protocols, focusing on security parameters and properties.
Pedagogical goals:
- generic cryptographic primitives: one-way, trap-door and hash functions; random generators; symmetric and assymertic cipher; interactive protocols;
- security properties : complexity and reduction proofs; undistinguidhability; non-malleability; soundness, completeness and zero-knowledge; confidentiality; authentication; privacy; non-repudiation
- use, deployment and integration of protocols in standard crypro lib (eg open-ssl)
- security proofs : fundations and verufucation based on tools (eg avispa)
UE From Basic Machine Learning models to Advanced Kernel Learning
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Statistical learning is about the construction and study of systems that can automatically learn from data. With the emergence of massive datasets commonly encountered today, the need for powerful machine learning is of acute importance. Examples of successful applications include effective web search, anti-spam software, computer vision, robotics, practical speech recognition, and a deeper understanding of the human genome. This course gives an introduction to this exciting field. In the first part, we will introduce basic techniques such as logistic regression, multilayer perceptrons, nearest neighbor approaches, both from a theoretical and methodological point of views. In the second part, we will focus on more advanced techniques such as kernel methods, which is a versatile tool to represent data, in combination with (un)supervised learning techniques that are agnostic to the type of data that is learned from. The learning techniques that will be covered include regression, classification, clustering and dimension reduction. We will cover both the theoretical underpinnings of kernels, as well as a series of kernels that are important in practical applications. Finally we will touch upon topics of active research, such as large-scale kernel methods and the use of kernel methods to develop theoretical foundations of deep learning models.
UE Mathematical Foundations of Machine Learning
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Machine Learning is one of the key areas of Artificial Intelligence and it concerns the study and the development of quantitative models that enables a computer to perform tasks without being explicitly programmed to do them. Learning in this context is hence to recognize complex forms and to make intelligent decisions. Given all existing entries, the difficulty of this task lies in the fact that all possible decisions is usually very complex to enumerate. To get around that, machine learning algorithms are designed in order to gain knowledge on the problem to be addressed based on a limited set of observed data extracted from this problem. To illustrate this principle, consider the supervised learning task, where the prediction function, which infers a predicted output for a given input, is learned over a finite set of labeled training examples, where each instance of this set is a pair constituted of a vector characterizing an observation in a given vector space, and an associated desired response for that instance (also called desired output). After the training step, the function returned by the algorithm is sought to give predictions on new examples, which have not been used in the learning process, with the lowest probability of error. The underlying assumption in this case is that the examples are, in general, representative of the prediction problem on which the function will be applied. We expect that the learning algorithm produces a function that will have a good generalization performance and not the one that is able to perfectly reproduce the outputs associated to the training examples. Guarantees of learnability of this process were studied in the theory of machine learning largely initiated by Vladimir Vapnik. These guarantees are dependent on the size of the training set and the complexity of the class of functions where the algorithm searches for the prediction function. Emerging technologies, particularly those related to the development of Internet, reshaped the domain of machine learning with new learning frameworks that have been studied to better tackle the related problems. One of these frameworks concerns the problem of learning with partially labeled data, or semi-supervised learning, which development is motivated by the effort that has to be made to construct labeled training sets for some problems, while large amount of unlabeled data can be gathered easily for these problems. The inherent assumption, in this case, is that unlabeled data contain relevant information about the task that has to be solved, and that it is a natural idea to try to extract this information so as to provide the learning algorithm more evidence. From these facts were born a number of works that intended to use a small amount of labeled data simultaneously with a large amount of unlabeled data to learn a prediction function.
The intent of this course is to propose a broad introduction to the field of Machine Learning, including discussions of each of the major frameworks, supervised, unsupervised, semi-supervised and reinforcement learning.
UE Learning, Probabilities and Causality
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Causality is at the core of our vision of the world and of the way we reason. It has long been recognized as an important concept and was already mentioned in the ancient Hindu scriptures: “Cause is the effect concealed, effect is the cause revealed”. Even Democritus famously proclaimed that he would rather discover a causal relation than be the king of presumably the wealthiest empire of his time. Nowadays, causality is seen as an ideal way to explain observed phenomena and to provide tools to reason on possible outcomes of interventions and what-if experiments, which are central to counterfactual reasoning, as ‘‘what if this patient had been given this particular treatment?’’
UE Statistical learning: from parametric to nonparametric models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is related to mathematical and statistical methods which are very used in supervised learning.
It contains two parts.
In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.
In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.
This course will be followed by practical sessions with the R software.
UE Mathematical optimization
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Safety Critical Systems: from design to verification
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées), Grenoble INP - Ensimag (Informatique, mathématiques appliquées et télécommunications), UGA
Semester
Automne
UE Natural Language Processing & Information Retrieval
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The automatic processing of languages, whether written or spoken, has always been an essential part of artificial intelligence. This domain has encouraged the emergence of new uses thanks to the arrival in the industrial field of many technologies from research (spell-checkers, speech synthesis, speech recognition, machine translation, …). In this course, we present the most recent advances and challenges for research. We will discuss discourse analysis whether written or spoken, text clarification, automatic speech transcription and automatic translation, in particular recent advances with neural models.
Information access and retrieval is now ubiquitous in everyday life through search engines, recommendation systems, or technological and commercial surveillance, in many application domains either general or specific like health for instance. In this course, we will cover Information retrieval basics, information retrieval evaluation, models for information retrieval, medical information retrieval, and deep learning for multimedia indexing and retrieval
UE Information Security
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This lecture deals with Information Systems Security and provides several facets, ranging from modeling to deployment in real applications. Information Systems Security refers to the processes and methodologies involved to keep information confidential, available, and assure its integrity. The lecture is divided in two major parts and assisted with several practical labs, allowing the students to model and configure security policies and also to be aware about several kinds of attacks and breaches.
Course content
Part 1: Access Control, or how to prevent unauthorized people from entering or accessing a system?
This part deals with:
- Cryptology for authentication and trust;
- security needs : confidentiality, availability, integrity, non-repudiation (CAI/DICP);
- cryptology primitives : private and public key ; trusted infrastructure (PKI and ledgers).
- zero-knowledge protocols; secret sharing and multiparty computation / or Practical work with Open SSL
- Access control mechanisms (MAC, DAC, RBAC, ABAC) and their implementations
- The detection and remediation of security breaches such as intrusions and insider attacks
- The deployment of control filters in applications and proxies.
The presented approach is built on the model-driven security paradigm (MDS). It refers to the process of modeling security requirements at a high level of abstraction, and generating technical security implementations. Security models are transformed into enforceable security rules including the run-time security management (e.g. entitlements/authorisations). Three labs are planned:
- B4MSecure: we will apply a formal language with animation and model-checking facilities to identify security breaches in Access Control policies.
- Snort: in this practical session you will set a local environment to simulate two machines, the target machine and the attacker. You will learn how to create firewall rules, monitor your network and how to react when an attack is detected.
- Metasploit: you will discover technology intelligence for vulnerabilities through a practical session where you will reproduce an exploit to hack and take control over a web-based server.
Part 2: Overview of modern attacks on systems, protocols, and networks and countermeasures
This part is devoted to modern attacks carried out on the Internet scale, in particular attacks on the DNS system (Domain Name System), such as cache or zone poisoning attacks, reflection and amplification of DDoS attacks (Distributed Denial of Service), IP spoofing - the root cause of DDoS attacks, botnets (e.g., Mirai), domain generation algorithms used for command-and-control communications, modern malware (e.g., Emotet trojan, Avalanche), spam, phishing, and business email compromise (BEC) scams.
The module will discuss preventative measures and security protocols to fight modern attacks, such as DDoS protection services, IP source address validation (SAV) known as BCP 38, Sender Policy Framework, and DMARC protocols as the first line of defense against email spoofing and BEC fraud, and DNSSEC to prevent DNS manipulation attacks. It will also discuss large-scale vulnerability measurements (a case study of the zone poisoning attack) and the challenges of deploying current security technologies by the system and network operators.
This part will be concluded with a practical team assignment in which students will be divided into groups and will have to configure a secure system in a real-world environment. The goal is to secure their system against the various types of discussed attacks and exploit other groups' systems.
UE Human Computer Interaction
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées), Grenoble INP - Ensimag (Informatique, mathématiques appliquées et télécommunications), UGA
Semester
Automne
UE Next Generation Software Development
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées), Grenoble INP - Ensimag (Informatique, mathématiques appliquées et télécommunications), UGA
Semester
Automne
UE Practicum
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Advanced models and methods in operations research
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course presents advanced methods and technics for Operations Research.
Reminder :
Linear Programming, Dynamic Programming, MIP modelling and BB
Complexity (P, NP, Co-NP)
Advanced MIP :
formulation, cuts, bounds
applications
lagragian relaxation
column generation
Benders decomposition
Solvers
Constraint Programming
Heuristics
local search
approximation algorithms
UE Combinatorial optimization and graph theory
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to provide a broad knowledge of fundamental problems in Combinatorial Optimization to show their algorithmic solutions and to derive min-max results on them. In order to achieve this goal a new object called a polyhedron is introduced. This polyhedral approach helps to shed new light on some classic results of Combinatorial Optimization.
Syllabus: Study of polyhedra associated to problems of Combinatorial Optimization ; Theory of blocking polyhedra ; Connectivity: shortest paths, spanning trees and spanning arborescences of minimum weight ; Flows: Edmonds-Karp Algorithm, Goldberg-Tarjan Algorithm, minimum cost flows ; Matchings: Hungarian method, Edmonds' Algorithm, Chinese postman problem; Matroids: greedy algorithm, intersection of two matroids ; Graph coloring ; Applications coming from various areas of Operations Research.
UE Optimization under uncertainty
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The objective of this course is to present different techniques to handle uncertainty in decision problems. These techniques will be illustrated on several applications e.g. inventory control, scheduling, energy, machine learning.
Syllabus : Introduction to uncertainty in optimization problems; Reminders (probability, dynamic programming, ...); Markov chains; Markov decision processes; Stochastic programming; Robust optimization
UE GS_MSTIC_Research ethics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Constraint Programming, applications in scheduling
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Graphs and discrete structures
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to learn how to use the structure of graphs and other discrete objects to obtain general results on them, and in particular efficient algorithms solving important problems.
We will cover the following topics:
- Structural Graph Theory: we will study the structure of important graph classes with nice algorithmic properties (planar graphs, interval graphs, comparability graphs, ...) and show several concrete problems that can be solved thanks to their structural properties.
- Graph Drawing: with the rise of Big Data, representing huge data sets is a fundamental challenge. Efficient ways to represent large graphs will be presented.
- Codes: we will see various codes (dominating, locating, identifying, ...) in graphs and their applications.
- Extremal combinatorics: the typical question in this field is "what global condition do we need to impose in some graph in order to make sure that some nice structure appears locally?" We will introduce a powerful tool called "the probabilistic method", an show how it can be applied to solve problems in this important area of research.
UE Advanced heuristic and approximation algorithms
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Advanced mathematical programming methods
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Academic and industrial challenges
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course offers the possibility for the students to gain some experience by facing open/difficult combinatorial problems.
The goal is to model and solve a combinatorial problem with direct industrial applications. We expect the students to take a variety of approaches (local search, compact/extended linear programming formulations, constraint programming, ...) and establish useful results (lower bounds, cuts, complexity,...).
The experimental results will be compared to the litterature (a known academic open benchmark will be available in this case) or will be validated by the industrial partner.
UE Transport Logistics and Operations Research
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Practicum
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course provides an introduction to the basic concepts of object-oriented programming in C++: classes and encapsulation, operator overloading, generic classes (templates), STL (Standard Template Library), inheritance and derived classes, polymorphism and virtual functions.
Lab sessions illustrate these concepts, and applications for applied mathematics.
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Connaitre les notions élémentaires et indispensables de probabilités. Maitriser les calculs de lois, d'espérance; connaitre les fonctions de simulation de logiciels spécialisés; savoir utiliser ces fonctions pour illustrer les résultats
de probabilité. Chaque notion sera abordée en cours et illustrée numériquement en séance
avec les étudiants.
Connaitre les principaux estimateurs, leurs propriétés.
Être capable de programmer les différents estimateurs dans des
situations simples; savoir illustrer par simulation les propriétés des estimateurs (biais, erreur quadratique).
Chaque notion sera abordée en cours et illustrée numériquement en séance avec les étudiants.
Contenu : Lois fondamentales discrètes et continues; Variables indépendantes, lois conditionnelles.
Espérance, espérance conditionnelle. Convergence de variables aléatoires. Théorème
centrale limite, loi des grands nombres. Vecteur gaussien, théorème Cochran. Notion
de chaine de Markov illustrée dans des TP.
Estimateur, estimation. Biais, erreur quadratique, convergence. Intervalle de
confiance. Estimateur des moindres carrés. Estimateur des moments. Estimateur du maximum
de vraisemblance. Equations estimantes.
UE Systèmes dynamiques
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce cours présente des manières géométriques de traiter et résoudre des problèmes décrits par des équations différentielles.
- chapitre I : Introduction : généralités sur les systèmes dynamiques
- chapitre II : Systèmes unidimensionnels : Les points fixes, linéarisation et stabilité, Exemple : le modèle logistique, Existence et unicité des solutions d'équations différentielles ordinaires
- chapitre III : Bifurcations : Bifurcation selle-nœud, Bifurcation transcritique, Bifurcation transcritique imparfaite, Bifurcation fourche, Bifurcation fourche supercritique, Bifurcation fourche sous-critique, Bifurcation fourche supercritique imparfaite
- chapitre IV : Champ de vecteur sur un cercle : Oscillateur uniforme, Oscillateur non-uniforme
- chapitre V : Flots bidimensionnels et applications : Existence et unicité des solutions et conséquences topologiques, Systèmes linéaires, Systèmes non-linéaires : linéarisation proche des points fixes, Cycles limites, Le théorème de Poincaré-Bendixson, Systèmes Liénard, Systèmes gradients, Fonctions de Liapunov
- chapitre VI : Bifurcations bidimensionnelles : Bifurcations selle-nœud, transcritique et fourche, Bifurcation de Hopf, Bifurcations globales de cycles
UE Instability and Turbulences
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
This course presents basic notions on instabilities and turbulence. We try to be as progressive as possible and to base our presentation on analyses of real experiments and real flows. We review few mathematical methods to analyze nonlinear systems in terms of instabilities. The students have to use their new knowledge to run and analyze numerical simulations of very simple systems. We then study some of the most important physical mechanisms for fluid instabilities and the corresponding criteria. We quickly present a zoology of common fluid instabilities and discuss the mechanisms and the possible technical implications. We give a broad introduction on turbulence and describe few fundamental methods and results, in particular the Richardson cascade, the Reynolds decomposition and the Kolmogorov spectra.
Teaching program:
1. General introduction
- Instabilities and turbulence, interest?
- Reynolds experiment and Reynolds number
- Incompressible Navier-Stokes equations: diffusion and advection
- An example: the wake of a cylinder
2. Instabilities and transition to turbulence
- Systems with few degrees of freedom
- Fluid instability mechanisms and conditions
- Other flows examples
3. Effects of variable density
- Boussinesq approximation
- Unstable stratification, Rayleigh-Taylor instability
- Rayleigh-Benard instability (Ra, Nu)
- Stable stratification, Kelvin-Helmoltz instability and Richardson number
4. Turbulence
- Introduction, Richardson cascade
- Average and Reynolds decomposition
- Experimental and numerical methods to study turbulence
- Statistical descriptions
For this course, the students have to write in LaTeX a report on their practical work. Thus, we spent some time for a first gentle introduction of this tool widely used in scientific academia.
UE Turbulence
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce module est une introduction à la turbulence phénoménologique et statistique. On s’intéresse aux définitions et propriétés de la turbulence en terme de processus physiques et leur description dans des familles types d’écoulements cisaillés que l’on peut retrouver dans la nature et en ingénierie.
jet turbulent
jet turbulent
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course provides an introduction to the basic concepts of object-oriented programming in C++: classes and encapsulation, operator overloading, generic classes (templates), STL (Standard Template Library), inheritance and derived classes, polymorphism and virtual functions.
Lab sessions illustrate these concepts, and applications for applied mathematics.
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Connaitre les notions élémentaires et indispensables de probabilités. Maitriser les calculs de lois, d'espérance; connaitre les fonctions de simulation de logiciels spécialisés; savoir utiliser ces fonctions pour illustrer les résultats
de probabilité. Chaque notion sera abordée en cours et illustrée numériquement en séance
avec les étudiants.
Connaitre les principaux estimateurs, leurs propriétés.
Être capable de programmer les différents estimateurs dans des
situations simples; savoir illustrer par simulation les propriétés des estimateurs (biais, erreur quadratique).
Chaque notion sera abordée en cours et illustrée numériquement en séance avec les étudiants.
Contenu : Lois fondamentales discrètes et continues; Variables indépendantes, lois conditionnelles.
Espérance, espérance conditionnelle. Convergence de variables aléatoires. Théorème
centrale limite, loi des grands nombres. Vecteur gaussien, théorème Cochran. Notion
de chaine de Markov illustrée dans des TP.
Estimateur, estimation. Biais, erreur quadratique, convergence. Intervalle de
confiance. Estimateur des moindres carrés. Estimateur des moments. Estimateur du maximum
de vraisemblance. Equations estimantes.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE GS_MSTIC_Scientific approach
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Algebra
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Holomorphic functions
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
- Holomorphic and analytical functions, in particular the equivalence between the two notions, exponential function and logarithm, principle of analytic continuation, principle of isolated zeros, Cauchy formula for the disc
- Elemental properties of holomorphic functions (Cauchy inequalities, sequences and series of holomorphic functions, property of the mean, and principle of the maximum)
- Cauchy theory (existence of primitives, Cauchy theorems)
- Meromorphic functions (classification of isolated singularities, meromorphic functions, residue theorem, Laurent series)
- Riemann conformal representation theorem
UE Probabilities
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Analysis
Level
Baccalaureate +4
ECTS
9 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Study and research work
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This Teaching Unit proposes a discovery of research in mathematics through the study of a subject describing a result or a mathematical theory, with which the student will have to familiarize themselves in order to appropriate them and to be able to account for them in a written report and an oral presentation.
In practice, a list of subjects is proposed during the first semester. Each student selects four subjects from this list, ranked from 1 to 4, and then the person responsible for the training assigns to each student a topic as far as possible among these four. As soon as the assignments are known, each student will contact the author of their subject, who will supervise them for this work throughout the second semester. Once the supervisor has presented the subject and the details of the expected work, the student meets them regularly to report on the progress of his/her work and progress.
The Supervised Research Work leads to the writing of a written report using LaTeX software, which must include an abstract and a bibliography, and an oral defence of 20 to 30 minutes, often followed by questions, before a jury which includes the supervisor. The report and the defence jointly contribute to the evaluation of the work carried out.
UE Effective algebra and cryptographie
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Compléments sur les EDP
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Compléments du premier semestre : principe du maximum faible pour les ÉDP elliptiques du second ordre, inégalité de Harnack
Compléments sur les espaces de Sobolev : injections de Sobolev, opérateurs d'extension, théorie des traces. Introduction aux distributions, distributions tempérées
Opérateurs maximaux monotones, théorème de Hille-Yosida
Équation de la chaleur dans Ω × ]0,+∞[, où Ω ⊂ ℝn est un domaine régulier : existence et unicité des solutions avec conditions au bord de Dirichlet et de Neumann ; principe du maximum pour les solutions de l'équation de la chaleur
Équation des ondes dans Ω × ]0,+∞[ : existence et unicité des solutions, propagation à vitesse finie
Équation de la chaleur semi-linéaire
UE Differential geometry
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to curves and surfaces.
Curves: Frénet references, covariant derivatives, reference fields, connection forms, structural equations.
Surfaces: surfaces of R^3, plane tangents, differential forms, differentiable applications between surfaces.
Abstract Variety: Whitney's Theorem
Curvature: Normal curvature, Gauss curvature, Geodesics. Case of surfaces of revolution.
Geometry of the surfaces of R^3: Egregium theorem, Gauss Bonnet's theorem
Possibility of complement: Transition to sub-varieties of Rn and abstract varieties
Possibility of complement: Introduction to dynamic systems. Vector fields and dynamic systems on varieties
UE Markov process
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Galois theory
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE English S8
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Opening UE (only if C1 level in English reached)
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Software security, secure programming and computer forensics
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The goal of this course is to enable students to acquire the following concepts:
- know how to identify the strengths/weaknesses of a programming language from the point of view of security;
- know the main causes and consequences of the usual software vulnerabilities
- know the protection mechanisms provided by the compilers, by the OS - understand the main techniques of code analysis for security (their interests, their limits)
UE Security architecture
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
- Introduction
- Motivation/Diffie-Hellman ; MitM ; Kerberos ;
- Electronic Signatures ; DSS ; RSA-PSS ;
- References : RFC/PKCS/FIPS
- Key Management.
- PKI elements, functions ; Certificates, ASN.1, X509, CRL ;
- Trust models
- PKIX : Administration ; migration ; OCSP, SCVP, Novomodo;
- Cross-certification ; Bridge ;
- Embedded Model : Certificates Browsers/OS; pinning, EV certs, notaries, bulletin board ;
- PGP + GnuPG ; Spooky/Sudsy ; IBE; CBE ;
- Authentification by PKI
- fips-196 and variants
- Key transport
- Authenticated Diffie-Hellman (SIGMA)
- TLS (handshake)
- Cybersecurity of industrial IT
- Electronic Signature and industrial PKI
- Certification and Security Policies
- PKI deployement in industry
- Attacks against certification authorities and similar services
- Evaluation Criteria and regulations (common criteria ; RGS ; e-IDAS)
- Application Security
- Transactions: EMV ; SET ; 3D-Secure ; bitcoin
- Messaging: E-mail, S/MIME ; OTR
- Web: https
- Threats
- Introduction / Concepts / Threat Landscape
- Network Architecture - Theats / Protection Layer 1 to 7
- Communication Security
- VPN: TLS, IPsec
- Firewall / proxying
- Wireless Security
- IPv6
- Routing: DNS / DNSSec ; TOR
- Canal: TLS ; IPsec
- OS Security
- hardening
- SeLinux, AppArmor, GRSec
- HIDS
UE Cryptographic engineering, protocols and security models, data privacy, coding and applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course present the main cryptographic primitives and security protocols, focusing on security parameters and properties.
Pedagogical goals:
- generic cryptographic primitives: one-way, trap-door and hash functions; random generators; symmetric and assymertic cipher; interactive protocols;
- security properties : complexity and reduction proofs; undistinguidhability; non-malleability; soundness, completeness and zero-knowledge; confidentiality; authentication; privacy; non-repudiation
- use, deployment and integration of protocols in standard crypro lib (eg open-ssl)
- security proofs : fundations and verufucation based on tools (eg avispa)
UE Threat and risk analysis, IT security audit and norms
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Concepts: Threats, Risks, Vulnerabilities. Introduction to Security assessment and audit methods and tools (ISO 27005, EBIOS, OSTMM)
UE Physical Security : Embedded, Smart Card, Quantum & Biometrics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Embedded systems: Principles of embedded systems design; smart cards, structure, and physical attacks; Design for Test and attacks on test structures; attacks through auxiliary channels; attacks by mistakes; countermeasures to the cited attacks.
Biometrics: objectives, fundamental principle, verification/authentication, various biometric modalities, examination of the most used modalities (fingerprint, facial recognition, iris) on the sensor side as well as on the algorithm side, the biometrics market, biometric performance evaluation (FAR & FRR), standardization, security of biometric systems (cryptography/vitality detection), introduction to encrypted biometry with cryptography (the grail of biometrics), protection of privacy, myths and realities.
Quantum: the postulates of quantum mechanics; how to use quantum information to make calculations, circuits and quantum algorithms; description of quantum information, density matrices, POVM measurements, fidelity, entropy; quantum error corrector codes; a bit of quantum communication complexity; use quantum information to make cryptography theoretically "secure", key exchange protocol BB84
UE Advanced cryptology
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to present some advanced topics in cryptography. The exact content may vary from one year to another; as an indication past topics have included:
- Linear secret sharing schemes (code-based schemes, access structures...)
- Provable constructions in symmetric cryptography (building block cipher from ideal permutations)
- Symmetric cryptanalysis (statistical and algebraic)
- Algorithms and constructions in code-based cryptography (information-set decoding, LPN)
- Zero-knowledge proofs
- Advanced signatures (group signatures...)
- Advanced constructions (oblivious transfer, group encryption...)
- Post-quantum cryptography
- Elliptic-curve and isogeny-based cryptography
UE Advanced security
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The advanced security module proposes to investigate deeper certain topics in security which include privacy models (k-anonymity, differential privacy and privacy by design), secure data structures (hash chain, Merkle's tree), (in)secure communication protocols (WEP and WPA protocols) and anti-viruses. The module focuses on several case study on privacy enhancing technologies (PETs), blockchain (along with an overview of cryptocurrencies), wireless attacks with scapy, malware detection using YARA and ClamAV.
UE Research practicum (in company or laboratory)
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
(en entreprise ou laboratoire)
UE Software security, secure programming and computer forensics
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The goal of this course is to enable students to acquire the following concepts:
- know how to identify the strengths/weaknesses of a programming language from the point of view of security;
- know the main causes and consequences of the usual software vulnerabilities
- know the protection mechanisms provided by the compilers, by the OS - understand the main techniques of code analysis for security (their interests, their limits)
UE Cryptographic engineering, protocols and security models, data privacy, coding and applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course present the main cryptographic primitives and security protocols, focusing on security parameters and properties.
Pedagogical goals:
- generic cryptographic primitives: one-way, trap-door and hash functions; random generators; symmetric and assymertic cipher; interactive protocols;
- security properties : complexity and reduction proofs; undistinguidhability; non-malleability; soundness, completeness and zero-knowledge; confidentiality; authentication; privacy; non-repudiation
- use, deployment and integration of protocols in standard crypro lib (eg open-ssl)
- security proofs : fundations and verufucation based on tools (eg avispa)
UE Threat and risk analysis, IT security audit and norms
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Concepts: Threats, Risks, Vulnerabilities. Introduction to Security assessment and audit methods and tools (ISO 27005, EBIOS, OSTMM)
UE Physical Security : Embedded, Smart Card, Quantum & Biometrics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Embedded systems: Principles of embedded systems design; smart cards, structure, and physical attacks; Design for Test and attacks on test structures; attacks through auxiliary channels; attacks by mistakes; countermeasures to the cited attacks.
Biometrics: objectives, fundamental principle, verification/authentication, various biometric modalities, examination of the most used modalities (fingerprint, facial recognition, iris) on the sensor side as well as on the algorithm side, the biometrics market, biometric performance evaluation (FAR & FRR), standardization, security of biometric systems (cryptography/vitality detection), introduction to encrypted biometry with cryptography (the grail of biometrics), protection of privacy, myths and realities.
Quantum: the postulates of quantum mechanics; how to use quantum information to make calculations, circuits and quantum algorithms; description of quantum information, density matrices, POVM measurements, fidelity, entropy; quantum error corrector codes; a bit of quantum communication complexity; use quantum information to make cryptography theoretically "secure", key exchange protocol BB84
UE GS_MSTIC_Research ethics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Advanced cryptology
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to present some advanced topics in cryptography. The exact content may vary from one year to another; as an indication past topics have included:
- Linear secret sharing schemes (code-based schemes, access structures...)
- Provable constructions in symmetric cryptography (building block cipher from ideal permutations)
- Symmetric cryptanalysis (statistical and algebraic)
- Algorithms and constructions in code-based cryptography (information-set decoding, LPN)
- Zero-knowledge proofs
- Advanced signatures (group signatures...)
- Advanced constructions (oblivious transfer, group encryption...)
- Post-quantum cryptography
- Elliptic-curve and isogeny-based cryptography
UE Advanced security
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The advanced security module proposes to investigate deeper certain topics in security which include privacy models (k-anonymity, differential privacy and privacy by design), secure data structures (hash chain, Merkle's tree), (in)secure communication protocols (WEP and WPA protocols) and anti-viruses. The module focuses on several case study on privacy enhancing technologies (PETs), blockchain (along with an overview of cryptocurrencies), wireless attacks with scapy, malware detection using YARA and ClamAV.
UE Research practicum (in company or laboratory)
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
(en entreprise ou laboratoire)
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
UE Systèmes dynamiques
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce cours présente des manières géométriques de traiter et résoudre des problèmes décrits par des équations différentielles.
- chapitre I : Introduction : généralités sur les systèmes dynamiques
- chapitre II : Systèmes unidimensionnels : Les points fixes, linéarisation et stabilité, Exemple : le modèle logistique, Existence et unicité des solutions d'équations différentielles ordinaires
- chapitre III : Bifurcations : Bifurcation selle-nœud, Bifurcation transcritique, Bifurcation transcritique imparfaite, Bifurcation fourche, Bifurcation fourche supercritique, Bifurcation fourche sous-critique, Bifurcation fourche supercritique imparfaite
- chapitre IV : Champ de vecteur sur un cercle : Oscillateur uniforme, Oscillateur non-uniforme
- chapitre V : Flots bidimensionnels et applications : Existence et unicité des solutions et conséquences topologiques, Systèmes linéaires, Systèmes non-linéaires : linéarisation proche des points fixes, Cycles limites, Le théorème de Poincaré-Bendixson, Systèmes Liénard, Systèmes gradients, Fonctions de Liapunov
- chapitre VI : Bifurcations bidimensionnelles : Bifurcations selle-nœud, transcritique et fourche, Bifurcation de Hopf, Bifurcations globales de cycles
UE Instability and Turbulences
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
This course presents basic notions on instabilities and turbulence. We try to be as progressive as possible and to base our presentation on analyses of real experiments and real flows. We review few mathematical methods to analyze nonlinear systems in terms of instabilities. The students have to use their new knowledge to run and analyze numerical simulations of very simple systems. We then study some of the most important physical mechanisms for fluid instabilities and the corresponding criteria. We quickly present a zoology of common fluid instabilities and discuss the mechanisms and the possible technical implications. We give a broad introduction on turbulence and describe few fundamental methods and results, in particular the Richardson cascade, the Reynolds decomposition and the Kolmogorov spectra.
Teaching program:
1. General introduction
- Instabilities and turbulence, interest?
- Reynolds experiment and Reynolds number
- Incompressible Navier-Stokes equations: diffusion and advection
- An example: the wake of a cylinder
2. Instabilities and transition to turbulence
- Systems with few degrees of freedom
- Fluid instability mechanisms and conditions
- Other flows examples
3. Effects of variable density
- Boussinesq approximation
- Unstable stratification, Rayleigh-Taylor instability
- Rayleigh-Benard instability (Ra, Nu)
- Stable stratification, Kelvin-Helmoltz instability and Richardson number
4. Turbulence
- Introduction, Richardson cascade
- Average and Reynolds decomposition
- Experimental and numerical methods to study turbulence
- Statistical descriptions
For this course, the students have to write in LaTeX a report on their practical work. Thus, we spent some time for a first gentle introduction of this tool widely used in scientific academia.
UE Turbulence
Level
Baccalaureate +4
ECTS
3 credits
Component
Faculté des sciences
Semester
Automne
Ce module est une introduction à la turbulence phénoménologique et statistique. On s’intéresse aux définitions et propriétés de la turbulence en terme de processus physiques et leur description dans des familles types d’écoulements cisaillés que l’on peut retrouver dans la nature et en ingénierie.
jet turbulent
jet turbulent
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Object-oriented and software design
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course provides an introduction to the basic concepts of object-oriented programming in C++: classes and encapsulation, operator overloading, generic classes (templates), STL (Standard Template Library), inheritance and derived classes, polymorphism and virtual functions.
Lab sessions illustrate these concepts, and applications for applied mathematics.
UE Partial differential equations and numerical methods
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Types of equations, conservation laws
- Finite differences methods
- Laplace equation
- Parabolic equations (diffusion)
- Hyperbolic equations (propagation)
- Non linear hyperbolic equations
This course include practical sessions.
This is a two parts course, this part is the course mutualized with Ensimag 2A 4MMMEDPS.
Partial differential equations and numerical methods
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Partial differential equations and numerical methods complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Signal and image processing
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Contents:
- Image definition
- Fourier transform, FFT, applications
- Image digitalisation, sampling
- Image processing: convolution, filtering. Applications
- Image decomposition, multiresolution. Application to compression
This course includes practical sessions.
UE Geometric modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is an introduction to the differential geometry of curves and surfaces with a particular focus on spline curves and surfaces that are routinely used in geometrical design softwares.
Content
This course includes practical sessions.
UE Applied probability and statistics
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Connaitre les notions élémentaires et indispensables de probabilités. Maitriser les calculs de lois, d'espérance; connaitre les fonctions de simulation de logiciels spécialisés; savoir utiliser ces fonctions pour illustrer les résultats
de probabilité. Chaque notion sera abordée en cours et illustrée numériquement en séance
avec les étudiants.
Connaitre les principaux estimateurs, leurs propriétés.
Être capable de programmer les différents estimateurs dans des
situations simples; savoir illustrer par simulation les propriétés des estimateurs (biais, erreur quadratique).
Chaque notion sera abordée en cours et illustrée numériquement en séance avec les étudiants.
Contenu : Lois fondamentales discrètes et continues; Variables indépendantes, lois conditionnelles.
Espérance, espérance conditionnelle. Convergence de variables aléatoires. Théorème
centrale limite, loi des grands nombres. Vecteur gaussien, théorème Cochran. Notion
de chaine de Markov illustrée dans des TP.
Estimateur, estimation. Biais, erreur quadratique, convergence. Intervalle de
confiance. Estimateur des moindres carrés. Estimateur des moments. Estimateur du maximum
de vraisemblance. Equations estimantes.
UE English
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The syllabus of the English course in M2 aims at enabling students to validate 3 competences that will be essential for their working life or for their doctoral studies in the future, at the B2 level:
1°) CAN give a clear presentation on a familiar topic, and CAN answer predictable or factual questions
2°) CAN find relevant information and essential points in written texts
3°) CAN make simple notes that are of reasonable use for essay or revision purposes, capturing most important points." (CEF, appendix D).
Reading comprehension can be validated in M1.
The course contents are linked to the students' fields of studies.
UE Computing science for big data and HPC
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Contents:
- Introduction to database
- Introduction to big data
- Introduction to high performance computing (HPC)
- Numerical solvers for HPC
HPC
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Introduction to database
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Project
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
January science and/or industrial project.
UE Internship
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Industrial and/or research internship.
UE Numerical optimisation
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.
Contents:
- Introduction, classification, examples.
- Theoretical results: convexity and compacity, optimality conditions, KT theorem
- Algorithmic for unconstrained optimisation (descent, line search, (quasi) Newton)
- Algorithms for non differentiable problems
- Algorithms for constrained optimisation: penalisatio, SQP methods
- Applications
UE GS_MSTIC_Scientific approach
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Operations Research
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Operational Research proposes scientific methods
to help make better decisions. The idea is to develop and use mathematical and computer
tools to master complex problems. Practical applications are historically in
the direction and management of large systems of people, machines, and
materials in industry, service, humanitarian aid, environment ...
In this course, we will focus on especially on problems with a combinatorial
structure: the number of possible solutions is finite but too large to be
enumerated. The study of these problems involves a phase of modelling practical
problems and then algorithmic resolution.
At the end of this course, students will be able to propose a model and will be
able to implement practical solutions (dedicated or industrial tools) to deal
with a problem of decision or optimization.
Operations Research Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Introduction to cryptology
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
To acquire the main theoretical and practical notions of modern cryptography: from notions in algorithmic complexity and information theory, to a general overview on the main algorithms and protocols in symmetric and asymmetric cryptography.
Introduction to cryptology complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D Graphics (AM)
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE 3D graphics
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Computer Graphics covers the set of techniques enabling the synthesis of animated virtual worlds. The applications range from entertainment (special effects, 3D feature films, video games), to industrial design (modelling and visualizing prototypes) and virtual reality (flight simulator, interactive walk-trough). This course introduces the domain by presenting the bases for the creation of 3D models, their animation, and the rendering of the corresponding 3D scene. Student will be invited to practice through programming exercises in OpenGL.
3D Graphics Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Turbulences
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Plasmas Astrophysiques et de Fusion
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
Experimental techniques in fluid mechanics
Level
Baccalaureate +4
Component
Faculté des sciences
Semester
Printemps
UE Variational methods applied to modelling
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Variational methods applied to modelling Complementary
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Statistical analysis and document mining
Level
Baccalaureate +4
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining
Level
Baccalaureate +4
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
Statistical analysis and document mining Complementary
Level
Baccalaureate +4
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
UE Differential Calculus, Wavelets and Applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is structured in two parts, treated respectively and independently by Sylvain Meignen and Kevin Polisano. The first part is devoted to differential calculus and its applications in image restoration and edge detection. The second part is dedicated to the construction and practical use of the wavelet transform. Wavelets are basis functions widely used in a large variety of fields: signal and image processing, data compression, smoothing/denoising data, numerical schemes for partial differential equations, scientific visualization, etc. Connections between the two parts will be made on the aspects of denoising, edge detection and graph analysis.
Course outline
Part I: Differential Calculus
- Differentiability on normed vector spaces
- Image restoration
- Edge detection
Part II: Wavelets and Applications
- From Fourier to the 1D Continuous Wavelet Transform
- Wavelet zoom, a local characterization of functions
- The 2D Continuous Wavelet Transform
- The 1D and 2D Discrete Wavelet Transform
- Linear and nonlinear approximations in wavelet bases
- The graph Fourier and wavelets transforms
UE An Introduction to Shape and Topology Optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In a very broad acceptation, shape and topology optimization is about finding the best domain (which may represent, depending on applications, a mechanical structure, a fluid channel,…) with respect to a given performance criterion (e.g. robustness, weight, etc.), under some constraints (e.g. of a geometric nature). Fostered by its impressive technological and industrial achievements, this discipline has aroused a growing enthusiasm among mathematicians, physicists and engineers since the seventies. Nowadays, problems pertaining to fields so diverse as mechanical engineering, fluid mechanics or biology, to name a few, are currently tackled with optimal design techniques, and constantly raise new, challenging issues.
UE Efficient methods in optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The subject of this half-semester course are more advanced methods in convex optimization. It consists of 6 lectures, 2 x 1,5 hours each, and can be seen as continuation of the course “Non-smooth methods in convex optimization”.
This course deals with:
Evaluation : A two-hours written exam (E1) in December. For those who do not pass there will be another two-hours exam (E2) in session 2 in spring.
UE Computational biology
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This interdisciplinary MSc course is designed for applicants with a biomedical, computational or mathematical background. It provides students with the necessary skills to produce effective research in bioinformatics and computational biology.
UE Fluid Mechanics and Granular Materials
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The first part of the lecture introduce to mathematical modeling of fluid mechanics and the numerical resolution of the associated equations. Equations are classified by three main families of models:
-
environmental problems: yield stress fluids (Bingham type) for granular matter, e.g. snow avalanches, mud or ice flows, erosion, landslides and volcanic lavas.
-
industrial problems: viscoelastic fluids (Oldroyd type) for plastic material processes, and metallic alloy.
-
biological problems: elastoviscoplastic fluids, for blood flows, liquid foam flows, and for food processing (mayonnaise, ketchup, etc).
Equations and models are presented in a continuum setting, and then approximated in time and space. Then, the efficient numerical resolution is addressed with some examples of practical applications.
The second part of the lecture propose a deeper analysis of granular models. The mathematical study of these complex matter is an important numerical and physical challenge. We will show how it requires a general view related to nonlinear PDEs. The objective of this course will be two-fold:
- Show how the compressibility and the viscoplasticity of the phenomenon can play an important role
- Discuss congestion phenomena in granular media (maximum packing) that can be compared mathematically to floating structure phenomena in the presence of a free boundary.
Students who complete the course will have demonstrated the ability to do the following:
- formulate and solve a large number of nonlinear physical and mechanical problems.
- demonstrate a familiarity with fluid mechanics and complex materials
- synthesize and implement efficient algorithms for various applications of industrial type.
The main idea of this lecture is to motivate by examples interdisciplinary collaborations needed to deal with complex situations.
UE GPU Computing
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In this course, we will introduce parallel programming paradigms to the students in the context of applied mathematics. The students will learn to identify the parallel pattern in numerical algorithm. The key components that the course will focus on are : efficiency, scalability, parallel pattern, comparison of parallel algorithms, operational intensity and emerging programming paradigm. Trough different lab assignments, the students will apply the concepts of efficient parallel programming using Graphic Processing Unit. In the final project, the students will have the possibility to parallelize one of their own numerical application developed in a previous course.
UE Software development tools and methods
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to study various useful applications, libraries and methods for software engineering related to applied mathematics. For example :
• C++ project management (git and/or svn)
• Development and profiling
• Boost library
• Linear algebra (Eigen)
• Prototyping and interfacing using Python
• Post processing and visualization tools (VTK, Paraview, GMSH)
This course deals with :
Topic 1: Software Engineering
Topic 2: Programming
Evaluation :
Practial sessions reports and oral presentation at the end of the course
UE Geophysical imaging
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In the current context of energy transition and fight against global warming, a precise knowledge of the crust, down to several km depth, has become a critical issue. The crust is the place where are to be found ore resources needed to build electric batteries (rare earth elements) as well as concrete resources for offshore and onshore wind turbines foundations. The crust is also the only place presenting sufficient volumes to store CO2 and H2 in a flexible way. CO2 storage will be a crucial component among industrial solutions to fight against global warming and reach neutral carbon emissions in the next decades.
To these ends, high resolution quantitative estimates of the mechanical parameters of the crust is essential. To perform such estimation, one has to rely on the interpretation of the mechanical waves which travel in the crust. The inference of the mechanical properties of the subsurface from local recording of the mechanical waves at the surface is a mathematical inverse problem. The aim of this course is to provide the mathematical background and the required theoretical tools to introduce high resolution seismic imaging methods to the students, complemented with practical numerical work on schematic examples.
The first main part of the course will be devoted to the theoretical and practical aspects of wave propagation in heterogeneous media. Beginning by some general consideration on hyperbolic partial differential equations, we will see how the elastodynamics equations, representing the propagation of mechanical waves in the subsurface, belong to this category of equations. We will show in particular an energy conservation result based on the symmetry of the underlying hyperbolic system. We will then discuss how to design absorbing boundary conditions for wave propagation problems, to mimic media of infinite extension. This will lead us to the question of numerical approximation to the solution of wave equations in heterogeneous media. We will discuss in details finite-difference schemes, and practical work will be dedicated to the implementation of a finite-difference scheme for the 1D and 2D acoustic equations, and potentially 2D elastic equations.
The second main part of the course will be devoted to the theoretical and practical aspects of seismic imaging using full waveform inversion. We will show how this method is formulated as a nonlinear inverse problem, controlled by partial differential equations representing wave propagation in heterogeneous media. We will discuss how this problem can be solved by local optimization strategies, and review such strategies, from 1st order gradient method to more evolved 2nd order Newton or quasi-Newton methods. The computation of the gradient of the misfit function through the adjoint state method, following optimal control theory, will be extensively presented, as well as its physical interpretation. This theoretical work will be supported by numerical experiments based on the finite-difference wave propagation code developed in the first part of the course. We will then discuss how full waveform inversion is applied in practice, supported by various field data applications examples. This will lead us to discuss current limitations of the method related to its ill-posedness and the lack of regularity of the solution, and give an overview of methodological work currently performed to mitigate these limitations.
Course outline
2 introductory session
- main introduction on seismic imaging (to do what?)
- main concepts related to general inverse problems
5 modeling sessions
- theoretical considerations on hyperbolic systems
- how to derive the elastodynamics equations from Newton and Hooke’s law
- elastodynamics equations = symmetrizable hyperbolic system, energy conservation
- absorbing boundary conditions - numerical approximation to the solution of wave propagation in heterogeneous media (finite-difference, finite element) - practical work : implement 1D and 2D acoustic, + 2D elastic if time allows
5 inverse problem sessions
- imaging the crust= nonlinear inverse problem controlled by an hyperbolic PDE
- local optimization method
- gradient computation through the adjoint state strategy
- physical interpretation of the gradient and Hessian operators - implementation of the gradient computation based on the modeling code designed in the first part
- full waveform inversion in practice: hierarchical schemes
- review of applications - review of current methodological developments
UE Handling uncertainties in (large-scale) numerical models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Numerical simulation is ubiquitous in today’s world. Initially confined to well-mastered physical problems, it has spread to all fields (oceanography, biology, ecology, etc.), the aim being to make forecasts of the systems under study. This has been possible thanks to the combination of numerical models and access to a considerable amount of data. However, there are many sources of uncertainty in these modelling systems. They can come from poorly known processes, approximations in the model equations and/or in their discretization, partial and uncertain data, … The objective of this course is to explore in depth the mathematical methods that have allowed these two worlds to meet. Firstly, we will focus on sensitivity analysis approaches that allow us to study the behavior of the system and its response to perturbations. In particular, this permits to study the way in which uncertainties are propagated. Next, we will look at data assimilation methods that aim at reducing said uncertainties by combining numerical models and observation data. Finally, the notions of model reduction will be discussed, which allow the implementation of the previous methods on high dimensional problems.
This course is intended for DS and MSCI students and will start with a differentiated refresher course on the necessary basic mathematical notions.
Course outline
-
General introduction and reminder of the basic concepts
-
Sensitivity analysis
-
Local sensitivity analysis
-
Global sensitivity analysis
-
-
Data assimilation
-
Variational methods
-
Stochastic methods
-
-
Model reduction
-
Gaussian processes
-
Polynomial Chaos
-
UE Modeling seminar and projects
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This lecture proposes modelling problems. The problems can be industrial or academic. Students are faced to an industrial problem or an academic problem (research oriented). They are in charge of this project. An teacher/tutor may guide them to find solutions to the problem. For industrial project, they have to understand the user needs, to analyze and model the problem, to derive specifications, to implement a solution and to develop the communication and the presentation of the proposed solution. More academic projects are linked to the courses. They are constructed such that the students can go deeper into a subject.
This lecture introduces basic communication methodes in industry. This part is in french and optional.
Rules: the students have to choose TWO subjects (either academic or industrial). They work in small groups on both projects with tutor (analysis of the problem, bibliography, construction of a solution, numerical simulations, etc.). At the end, they defend their results in front of a jury and provide a short report.
UE Quantum Information & Dynamics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The quantum formalism developed a century ago provides a very precise description of nature at small scales which entails several counter intuitive aspects: superposition of states, entanglement, intrinsic randomness of measurement process, to list a few. However, from a mathematical point of view, quantum mechanics does have a definite formulation. This allows to investigate these intriguing features rigorously and to explore these quantum traits in information theory and algorithmics in particular, as well as the challenges they present.
The goal of these lectures is to provide a mathematical description of the quantum formalism in finite dimension and to introduce the mathematical concepts and tools required for the analysis of such quantum systems and their dynamics. On the one hand, we will study the key aspects of quantum information theory. On the other hand, we will describe certain properties of quantum dynamics that need to be taken into account in the implementation of quantum algorithms and that will be applied to emblematic systems. The interaction with an external classical electromagnetic field will also be considered both from a theoretical and a numerical point of view.
UE Optimal transport: theory, applications and related numerical methods
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The goal of this course is to present a wide range of recent numerical methods and algorithms that find applications in various fields. More precisely, the course will focus on optimal transport algorithms, proximal methods and level set methods – the leading application of these being image analysis.
Optimal transport is an important field of mathematics that was originally introduced in the 1700’s by the French mathematician and engineer Gaspard Monge to answer the following concrete question: what is the cheapest way of sending a pile of sand into a hole, knowing the cost of transportation of each sand grain of the pile to a possible target location? This problem gave the birth of the theory of optimal transport. This theory has connections with PDEs, geometry and probability and has been used in many fields such as computer vision, economy, non-imaging optics… In the last 15 years, this problem has been extensively studied from a computational point of view and different efficient algorithms have been proposed.
In this course, we will provide an introduction to the optimal transport theory and the analysis of several algorithms dedicated to the discrete case (i.e. when the source and target measures are discrete), such as Auction’s algorithm and Sinkorn algorithm. We will also study the semi-discrete setting that corresponds to transporting a continuous measure to a discrete one and we will analyze Oliker-Prüssner algorithm as well as a Newton algorithm.
We will also consider some numerical methods which have wide applications in several modeling fields as the Level Set method to capture interfaces, primal dual methods, with main application in this course to image analysis : active contours, deblurring, demonising, inpainting and interpolation, the latter issue being dealt with by a so-called dynamic formulation of optimal transport.
UE Statistical learning: from parametric to nonparametric models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is related to mathematical and statistical methods which are very used in supervised learning.
It contains two parts.
In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.
In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.
This course will be followed by practical sessions with the R software.
UE Temporal, spatial and extreme event analysis
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Modelling extreme temperatures, extreme river flows, earthquakes intensities, neuronal activity, map diseases, lightning strikes, forest fires, for example is a risk modelling and assessment task, which is tackled in statistics using point processes and extreme value theory.
On the one hand, point processes are a class of stochastic processes modelling random events in interaction. By event we can think of the time a neuron activates, an earthquake occurs, the time a tweet has been retweeted, etc or the location of a tree in a forest, the impact of a lightning strike, etc. The first two parts provide an introduction to stochastic models and statistical inference which could cover such applications. Main characteristics of such processes, standard models (properties, simulation) and statistical procedures to infer them will be presented.
On the other hand, taking into account extreme events such as heavy rainfalls, floods, extreme temperatures is often crucial in the statistical approach to risk modeling. In this context, the behavior of the distribution tail is then more important than the shape of the central part of the distribution. Extreme-value theory offers a wide range of tools for modeling and estimating the probability of extreme events.
UE Research projects
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This is the master thesis project.
UE Advanced Machine Learning: Applications to Vision, Audio and Text
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is split into two parts. During the first part, a wide range of machine learning algorithms will be discussed. The second part will focus on deep learning, and presentations more applied to the three data modalities and their combinations. The following is a non-exhaustive list of topics discussed:
- Computing dot products in high dimension & Page Rank
- Matrix completion/factorization (Stochastic Gradient Descent, SVD)
- Monte-carlo, MCMC methods: Metropolis-Hastings and Gibbs Sampling
- Unsupervised classification: Partitionning, Hierarchical, Kernel and Spectral clustering
- Alignment and matching algorithms (local/global, pairwise/multiple), dynamic programming, Hungarian algorithm,…
- Introduction to Deep Learning concepts, including CNN, RNN, Metric learning
- Attention models: Self-attention, Transformers
- Auditory data: Representation, sound source localisation and separation.
- Natural language data: Representation, Seq2Seq, Word2Vec, Machine Translation, Pre-training strategies, Benchmarks and evaluation
- Visual data: image and video representation, recap of traditional features, state-of-the-art neural architectures for feature extraction
- Object detection and recognition, action recognition.
- Multimodal learning: audio-visual data representation, multimedia retrieval.
- Generative Adversarial Networks: Image-image translation, conditional generation
UE An Introduction to Shape and Topology Optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In a very broad acceptation, shape and topology optimization is about finding the best domain (which may represent, depending on applications, a mechanical structure, a fluid channel,…) with respect to a given performance criterion (e.g. robustness, weight, etc.), under some constraints (e.g. of a geometric nature). Fostered by its impressive technological and industrial achievements, this discipline has aroused a growing enthusiasm among mathematicians, physicists and engineers since the seventies. Nowadays, problems pertaining to fields so diverse as mechanical engineering, fluid mechanics or biology, to name a few, are currently tackled with optimal design techniques, and constantly raise new, challenging issues.
UE Computational biology
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This interdisciplinary MSc course is designed for applicants with a biomedical, computational or mathematical background. It provides students with the necessary skills to produce effective research in bioinformatics and computational biology.
UE Data Science Seminars and Challenge
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course contains two parts.
Part I concerns Data challenge.
This part consists in a real problem that is given to the students for which data are readily available. The goal is to have teams of five to six students compete in solving (at least partially) the problem.
The work is spread over the Autumn semester and consists of: building a prediction model or a methodology to solve the problem based on a set of training data, blind evaluation of the model or methodology on a test bench (unseen data, withheld from the students), using an appropriate performance measure.
At the end, the teams will present their solution path in a formal presentation and a short report.
Part II concerns Data Science seminars.
This is a cycle of seminars or presentations with a common factor that is the project of the data challenge. A first seminar will settle the context and the problem for that year’s data challenge.
The other seminars will propose different industrial or academic approaches and problems that are (loosely) related to the objective of the data challenge. Presentations have a time slot of one hour and students will have to read up front some ressources to orient their questions about the subject after the seminar.
UE Differential Calculus, Wavelets and Applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is structured in two parts, treated respectively and independently by Sylvain Meignen and Kevin Polisano. The first part is devoted to differential calculus and its applications in image restoration and edge detection. The second part is dedicated to the construction and practical use of the wavelet transform. Wavelets are basis functions widely used in a large variety of fields: signal and image processing, data compression, smoothing/denoising data, numerical schemes for partial differential equations, scientific visualization, etc. Connections between the two parts will be made on the aspects of denoising, edge detection and graph analysis.
Course outline
Part I: Differential Calculus
- Differentiability on normed vector spaces
- Image restoration
- Edge detection
Part II: Wavelets and Applications
- From Fourier to the 1D Continuous Wavelet Transform
- Wavelet zoom, a local characterization of functions
- The 2D Continuous Wavelet Transform
- The 1D and 2D Discrete Wavelet Transform
- Linear and nonlinear approximations in wavelet bases
- The graph Fourier and wavelets transforms
UE Efficient methods in optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The subject of this half-semester course are more advanced methods in convex optimization. It consists of 6 lectures, 2 x 1,5 hours each, and can be seen as continuation of the course “Non-smooth methods in convex optimization”.
This course deals with:
Evaluation : A two-hours written exam (E1) in December. For those who do not pass there will be another two-hours exam (E2) in session 2 in spring.
UE From Basic Machine Learning models to Advanced Kernel Learning
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Statistical learning is about the construction and study of systems that can automatically learn from data. With the emergence of massive datasets commonly encountered today, the need for powerful machine learning is of acute importance. Examples of successful applications include effective web search, anti-spam software, computer vision, robotics, practical speech recognition, and a deeper understanding of the human genome. This course gives an introduction to this exciting field. In the first part, we will introduce basic techniques such as logistic regression, multilayer perceptrons, nearest neighbor approaches, both from a theoretical and methodological point of views. In the second part, we will focus on more advanced techniques such as kernel methods, which is a versatile tool to represent data, in combination with (un)supervised learning techniques that are agnostic to the type of data that is learned from. The learning techniques that will be covered include regression, classification, clustering and dimension reduction. We will cover both the theoretical underpinnings of kernels, as well as a series of kernels that are important in practical applications. Finally we will touch upon topics of active research, such as large-scale kernel methods and the use of kernel methods to develop theoretical foundations of deep learning models.
UE Handling uncertainties in (large-scale) numerical models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Numerical simulation is ubiquitous in today’s world. Initially confined to well-mastered physical problems, it has spread to all fields (oceanography, biology, ecology, etc.), the aim being to make forecasts of the systems under study. This has been possible thanks to the combination of numerical models and access to a considerable amount of data. However, there are many sources of uncertainty in these modelling systems. They can come from poorly known processes, approximations in the model equations and/or in their discretization, partial and uncertain data, … The objective of this course is to explore in depth the mathematical methods that have allowed these two worlds to meet. Firstly, we will focus on sensitivity analysis approaches that allow us to study the behavior of the system and its response to perturbations. In particular, this permits to study the way in which uncertainties are propagated. Next, we will look at data assimilation methods that aim at reducing said uncertainties by combining numerical models and observation data. Finally, the notions of model reduction will be discussed, which allow the implementation of the previous methods on high dimensional problems.
This course is intended for DS and MSCI students and will start with a differentiated refresher course on the necessary basic mathematical notions.
Course outline
-
General introduction and reminder of the basic concepts
-
Sensitivity analysis
-
Local sensitivity analysis
-
Global sensitivity analysis
-
-
Data assimilation
-
Variational methods
-
Stochastic methods
-
-
Model reduction
-
Gaussian processes
-
Polynomial Chaos
-
UE GPU Computing
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In this course, we will introduce parallel programming paradigms to the students in the context of applied mathematics. The students will learn to identify the parallel pattern in numerical algorithm. The key components that the course will focus on are : efficiency, scalability, parallel pattern, comparison of parallel algorithms, operational intensity and emerging programming paradigm. Trough different lab assignments, the students will apply the concepts of efficient parallel programming using Graphic Processing Unit. In the final project, the students will have the possibility to parallelize one of their own numerical application developed in a previous course.
UE Learning, Probabilities and Causality
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Causality is at the core of our vision of the world and of the way we reason. It has long been recognized as an important concept and was already mentioned in the ancient Hindu scriptures: “Cause is the effect concealed, effect is the cause revealed”. Even Democritus famously proclaimed that he would rather discover a causal relation than be the king of presumably the wealthiest empire of his time. Nowadays, causality is seen as an ideal way to explain observed phenomena and to provide tools to reason on possible outcomes of interventions and what-if experiments, which are central to counterfactual reasoning, as ‘‘what if this patient had been given this particular treatment?’’
UE Mathematical Foundations of Machine Learning
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Machine Learning is one of the key areas of Artificial Intelligence and it concerns the study and the development of quantitative models that enables a computer to perform tasks without being explicitly programmed to do them. Learning in this context is hence to recognize complex forms and to make intelligent decisions. Given all existing entries, the difficulty of this task lies in the fact that all possible decisions is usually very complex to enumerate. To get around that, machine learning algorithms are designed in order to gain knowledge on the problem to be addressed based on a limited set of observed data extracted from this problem. To illustrate this principle, consider the supervised learning task, where the prediction function, which infers a predicted output for a given input, is learned over a finite set of labeled training examples, where each instance of this set is a pair constituted of a vector characterizing an observation in a given vector space, and an associated desired response for that instance (also called desired output). After the training step, the function returned by the algorithm is sought to give predictions on new examples, which have not been used in the learning process, with the lowest probability of error. The underlying assumption in this case is that the examples are, in general, representative of the prediction problem on which the function will be applied. We expect that the learning algorithm produces a function that will have a good generalization performance and not the one that is able to perfectly reproduce the outputs associated to the training examples. Guarantees of learnability of this process were studied in the theory of machine learning largely initiated by Vladimir Vapnik. These guarantees are dependent on the size of the training set and the complexity of the class of functions where the algorithm searches for the prediction function. Emerging technologies, particularly those related to the development of Internet, reshaped the domain of machine learning with new learning frameworks that have been studied to better tackle the related problems. One of these frameworks concerns the problem of learning with partially labeled data, or semi-supervised learning, which development is motivated by the effort that has to be made to construct labeled training sets for some problems, while large amount of unlabeled data can be gathered easily for these problems. The inherent assumption, in this case, is that unlabeled data contain relevant information about the task that has to be solved, and that it is a natural idea to try to extract this information so as to provide the learning algorithm more evidence. From these facts were born a number of works that intended to use a small amount of labeled data simultaneously with a large amount of unlabeled data to learn a prediction function.
The intent of this course is to propose a broad introduction to the field of Machine Learning, including discussions of each of the major frameworks, supervised, unsupervised, semi-supervised and reinforcement learning.
UE Modeling seminar and projects
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This lecture proposes modelling problems. The problems can be industrial or academic. Students are faced to an industrial problem or an academic problem (research oriented). They are in charge of this project. An teacher/tutor may guide them to find solutions to the problem. For industrial project, they have to understand the user needs, to analyze and model the problem, to derive specifications, to implement a solution and to develop the communication and the presentation of the proposed solution. More academic projects are linked to the courses. They are constructed such that the students can go deeper into a subject.
This lecture introduces basic communication methodes in industry. This part is in french and optional.
Rules: the students have to choose TWO subjects (either academic or industrial). They work in small groups on both projects with tutor (analysis of the problem, bibliography, construction of a solution, numerical simulations, etc.). At the end, they defend their results in front of a jury and provide a short report.
UE Optimal transport: theory, applications and related numerical methods
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The goal of this course is to present a wide range of recent numerical methods and algorithms that find applications in various fields. More precisely, the course will focus on optimal transport algorithms, proximal methods and level set methods – the leading application of these being image analysis.
Optimal transport is an important field of mathematics that was originally introduced in the 1700’s by the French mathematician and engineer Gaspard Monge to answer the following concrete question: what is the cheapest way of sending a pile of sand into a hole, knowing the cost of transportation of each sand grain of the pile to a possible target location? This problem gave the birth of the theory of optimal transport. This theory has connections with PDEs, geometry and probability and has been used in many fields such as computer vision, economy, non-imaging optics… In the last 15 years, this problem has been extensively studied from a computational point of view and different efficient algorithms have been proposed.
In this course, we will provide an introduction to the optimal transport theory and the analysis of several algorithms dedicated to the discrete case (i.e. when the source and target measures are discrete), such as Auction’s algorithm and Sinkorn algorithm. We will also study the semi-discrete setting that corresponds to transporting a continuous measure to a discrete one and we will analyze Oliker-Prüssner algorithm as well as a Newton algorithm.
We will also consider some numerical methods which have wide applications in several modeling fields as the Level Set method to capture interfaces, primal dual methods, with main application in this course to image analysis : active contours, deblurring, demonising, inpainting and interpolation, the latter issue being dealt with by a so-called dynamic formulation of optimal transport.
UE Natural Language Processing & Information Retrieval
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The automatic processing of languages, whether written or spoken, has always been an essential part of artificial intelligence. This domain has encouraged the emergence of new uses thanks to the arrival in the industrial field of many technologies from research (spell-checkers, speech synthesis, speech recognition, machine translation, …). In this course, we present the most recent advances and challenges for research. We will discuss discourse analysis whether written or spoken, text clarification, automatic speech transcription and automatic translation, in particular recent advances with neural models.
Information access and retrieval is now ubiquitous in everyday life through search engines, recommendation systems, or technological and commercial surveillance, in many application domains either general or specific like health for instance. In this course, we will cover Information retrieval basics, information retrieval evaluation, models for information retrieval, medical information retrieval, and deep learning for multimedia indexing and retrieval
UE Statistical learning: from parametric to nonparametric models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is related to mathematical and statistical methods which are very used in supervised learning.
It contains two parts.
In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.
In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.
This course will be followed by practical sessions with the R software.
UE Software development tools and methods
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to study various useful applications, libraries and methods for software engineering related to applied mathematics. For example :
• C++ project management (git and/or svn)
• Development and profiling
• Boost library
• Linear algebra (Eigen)
• Prototyping and interfacing using Python
• Post processing and visualization tools (VTK, Paraview, GMSH)
This course deals with :
Topic 1: Software Engineering
Topic 2: Programming
Evaluation :
Practial sessions reports and oral presentation at the end of the course
UE Temporal, spatial and extreme event analysis
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Modelling extreme temperatures, extreme river flows, earthquakes intensities, neuronal activity, map diseases, lightning strikes, forest fires, for example is a risk modelling and assessment task, which is tackled in statistics using point processes and extreme value theory.
On the one hand, point processes are a class of stochastic processes modelling random events in interaction. By event we can think of the time a neuron activates, an earthquake occurs, the time a tweet has been retweeted, etc or the location of a tree in a forest, the impact of a lightning strike, etc. The first two parts provide an introduction to stochastic models and statistical inference which could cover such applications. Main characteristics of such processes, standard models (properties, simulation) and statistical procedures to infer them will be presented.
On the other hand, taking into account extreme events such as heavy rainfalls, floods, extreme temperatures is often crucial in the statistical approach to risk modeling. In this context, the behavior of the distribution tail is then more important than the shape of the central part of the distribution. Extreme-value theory offers a wide range of tools for modeling and estimating the probability of extreme events.
UE Research projects
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This is the master thesis project.
UE GS_MSTIC_Research ethics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
UE Software development tools and methods
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The aim of this course is to study various useful applications, libraries and methods for software engineering related to applied mathematics. For example :
• C++ project management (git and/or svn)
• Development and profiling
• Boost library
• Linear algebra (Eigen)
• Prototyping and interfacing using Python
• Post processing and visualization tools (VTK, Paraview, GMSH)
This course deals with :
Topic 1: Software Engineering
Topic 2: Programming
Evaluation :
Practial sessions reports and oral presentation at the end of the course
UE Modeling seminar and projects
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This lecture proposes modelling problems. The problems can be industrial or academic. Students are faced to an industrial problem or an academic problem (research oriented). They are in charge of this project. An teacher/tutor may guide them to find solutions to the problem. For industrial project, they have to understand the user needs, to analyze and model the problem, to derive specifications, to implement a solution and to develop the communication and the presentation of the proposed solution. More academic projects are linked to the courses. They are constructed such that the students can go deeper into a subject.
This lecture introduces basic communication methodes in industry. This part is in french and optional.
Rules: the students have to choose TWO subjects (either academic or industrial). They work in small groups on both projects with tutor (analysis of the problem, bibliography, construction of a solution, numerical simulations, etc.). At the end, they defend their results in front of a jury and provide a short report.
UE Geophysical imaging
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In the current context of energy transition and fight against global warming, a precise knowledge of the crust, down to several km depth, has become a critical issue. The crust is the place where are to be found ore resources needed to build electric batteries (rare earth elements) as well as concrete resources for offshore and onshore wind turbines foundations. The crust is also the only place presenting sufficient volumes to store CO2 and H2 in a flexible way. CO2 storage will be a crucial component among industrial solutions to fight against global warming and reach neutral carbon emissions in the next decades.
To these ends, high resolution quantitative estimates of the mechanical parameters of the crust is essential. To perform such estimation, one has to rely on the interpretation of the mechanical waves which travel in the crust. The inference of the mechanical properties of the subsurface from local recording of the mechanical waves at the surface is a mathematical inverse problem. The aim of this course is to provide the mathematical background and the required theoretical tools to introduce high resolution seismic imaging methods to the students, complemented with practical numerical work on schematic examples.
The first main part of the course will be devoted to the theoretical and practical aspects of wave propagation in heterogeneous media. Beginning by some general consideration on hyperbolic partial differential equations, we will see how the elastodynamics equations, representing the propagation of mechanical waves in the subsurface, belong to this category of equations. We will show in particular an energy conservation result based on the symmetry of the underlying hyperbolic system. We will then discuss how to design absorbing boundary conditions for wave propagation problems, to mimic media of infinite extension. This will lead us to the question of numerical approximation to the solution of wave equations in heterogeneous media. We will discuss in details finite-difference schemes, and practical work will be dedicated to the implementation of a finite-difference scheme for the 1D and 2D acoustic equations, and potentially 2D elastic equations.
The second main part of the course will be devoted to the theoretical and practical aspects of seismic imaging using full waveform inversion. We will show how this method is formulated as a nonlinear inverse problem, controlled by partial differential equations representing wave propagation in heterogeneous media. We will discuss how this problem can be solved by local optimization strategies, and review such strategies, from 1st order gradient method to more evolved 2nd order Newton or quasi-Newton methods. The computation of the gradient of the misfit function through the adjoint state method, following optimal control theory, will be extensively presented, as well as its physical interpretation. This theoretical work will be supported by numerical experiments based on the finite-difference wave propagation code developed in the first part of the course. We will then discuss how full waveform inversion is applied in practice, supported by various field data applications examples. This will lead us to discuss current limitations of the method related to its ill-posedness and the lack of regularity of the solution, and give an overview of methodological work currently performed to mitigate these limitations.
Course outline
2 introductory session
- main introduction on seismic imaging (to do what?)
- main concepts related to general inverse problems
5 modeling sessions
- theoretical considerations on hyperbolic systems
- how to derive the elastodynamics equations from Newton and Hooke’s law
- elastodynamics equations = symmetrizable hyperbolic system, energy conservation
- absorbing boundary conditions - numerical approximation to the solution of wave propagation in heterogeneous media (finite-difference, finite element) - practical work : implement 1D and 2D acoustic, + 2D elastic if time allows
5 inverse problem sessions
- imaging the crust= nonlinear inverse problem controlled by an hyperbolic PDE
- local optimization method
- gradient computation through the adjoint state strategy
- physical interpretation of the gradient and Hessian operators - implementation of the gradient computation based on the modeling code designed in the first part
- full waveform inversion in practice: hierarchical schemes
- review of applications - review of current methodological developments
UE An Introduction to Shape and Topology Optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In a very broad acceptation, shape and topology optimization is about finding the best domain (which may represent, depending on applications, a mechanical structure, a fluid channel,…) with respect to a given performance criterion (e.g. robustness, weight, etc.), under some constraints (e.g. of a geometric nature). Fostered by its impressive technological and industrial achievements, this discipline has aroused a growing enthusiasm among mathematicians, physicists and engineers since the seventies. Nowadays, problems pertaining to fields so diverse as mechanical engineering, fluid mechanics or biology, to name a few, are currently tackled with optimal design techniques, and constantly raise new, challenging issues.
UE Refresh courses
Level
Baccalaureate +5
ECTS
0 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Refresh courses
UE GPU Computing
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
In this course, we will introduce parallel programming paradigms to the students in the context of applied mathematics. The students will learn to identify the parallel pattern in numerical algorithm. The key components that the course will focus on are : efficiency, scalability, parallel pattern, comparison of parallel algorithms, operational intensity and emerging programming paradigm. Trough different lab assignments, the students will apply the concepts of efficient parallel programming using Graphic Processing Unit. In the final project, the students will have the possibility to parallelize one of their own numerical application developed in a previous course.
UE Differential Calculus, Wavelets and Applications
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is structured in two parts, treated respectively and independently by Sylvain Meignen and Kevin Polisano. The first part is devoted to differential calculus and its applications in image restoration and edge detection. The second part is dedicated to the construction and practical use of the wavelet transform. Wavelets are basis functions widely used in a large variety of fields: signal and image processing, data compression, smoothing/denoising data, numerical schemes for partial differential equations, scientific visualization, etc. Connections between the two parts will be made on the aspects of denoising, edge detection and graph analysis.
Course outline
Part I: Differential Calculus
- Differentiability on normed vector spaces
- Image restoration
- Edge detection
Part II: Wavelets and Applications
- From Fourier to the 1D Continuous Wavelet Transform
- Wavelet zoom, a local characterization of functions
- The 2D Continuous Wavelet Transform
- The 1D and 2D Discrete Wavelet Transform
- Linear and nonlinear approximations in wavelet bases
- The graph Fourier and wavelets transforms
UE Optimal transport: theory, applications and related numerical methods
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The goal of this course is to present a wide range of recent numerical methods and algorithms that find applications in various fields. More precisely, the course will focus on optimal transport algorithms, proximal methods and level set methods – the leading application of these being image analysis.
Optimal transport is an important field of mathematics that was originally introduced in the 1700’s by the French mathematician and engineer Gaspard Monge to answer the following concrete question: what is the cheapest way of sending a pile of sand into a hole, knowing the cost of transportation of each sand grain of the pile to a possible target location? This problem gave the birth of the theory of optimal transport. This theory has connections with PDEs, geometry and probability and has been used in many fields such as computer vision, economy, non-imaging optics… In the last 15 years, this problem has been extensively studied from a computational point of view and different efficient algorithms have been proposed.
In this course, we will provide an introduction to the optimal transport theory and the analysis of several algorithms dedicated to the discrete case (i.e. when the source and target measures are discrete), such as Auction’s algorithm and Sinkorn algorithm. We will also study the semi-discrete setting that corresponds to transporting a continuous measure to a discrete one and we will analyze Oliker-Prüssner algorithm as well as a Newton algorithm.
We will also consider some numerical methods which have wide applications in several modeling fields as the Level Set method to capture interfaces, primal dual methods, with main application in this course to image analysis : active contours, deblurring, demonising, inpainting and interpolation, the latter issue being dealt with by a so-called dynamic formulation of optimal transport.
UE Fluid Mechanics and Granular Materials
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The first part of the lecture introduce to mathematical modeling of fluid mechanics and the numerical resolution of the associated equations. Equations are classified by three main families of models:
-
environmental problems: yield stress fluids (Bingham type) for granular matter, e.g. snow avalanches, mud or ice flows, erosion, landslides and volcanic lavas.
-
industrial problems: viscoelastic fluids (Oldroyd type) for plastic material processes, and metallic alloy.
-
biological problems: elastoviscoplastic fluids, for blood flows, liquid foam flows, and for food processing (mayonnaise, ketchup, etc).
Equations and models are presented in a continuum setting, and then approximated in time and space. Then, the efficient numerical resolution is addressed with some examples of practical applications.
The second part of the lecture propose a deeper analysis of granular models. The mathematical study of these complex matter is an important numerical and physical challenge. We will show how it requires a general view related to nonlinear PDEs. The objective of this course will be two-fold:
- Show how the compressibility and the viscoplasticity of the phenomenon can play an important role
- Discuss congestion phenomena in granular media (maximum packing) that can be compared mathematically to floating structure phenomena in the presence of a free boundary.
Students who complete the course will have demonstrated the ability to do the following:
- formulate and solve a large number of nonlinear physical and mechanical problems.
- demonstrate a familiarity with fluid mechanics and complex materials
- synthesize and implement efficient algorithms for various applications of industrial type.
The main idea of this lecture is to motivate by examples interdisciplinary collaborations needed to deal with complex situations.
UE Handling uncertainties in (large-scale) numerical models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Numerical simulation is ubiquitous in today’s world. Initially confined to well-mastered physical problems, it has spread to all fields (oceanography, biology, ecology, etc.), the aim being to make forecasts of the systems under study. This has been possible thanks to the combination of numerical models and access to a considerable amount of data. However, there are many sources of uncertainty in these modelling systems. They can come from poorly known processes, approximations in the model equations and/or in their discretization, partial and uncertain data, … The objective of this course is to explore in depth the mathematical methods that have allowed these two worlds to meet. Firstly, we will focus on sensitivity analysis approaches that allow us to study the behavior of the system and its response to perturbations. In particular, this permits to study the way in which uncertainties are propagated. Next, we will look at data assimilation methods that aim at reducing said uncertainties by combining numerical models and observation data. Finally, the notions of model reduction will be discussed, which allow the implementation of the previous methods on high dimensional problems.
This course is intended for DS and MSCI students and will start with a differentiated refresher course on the necessary basic mathematical notions.
Course outline
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General introduction and reminder of the basic concepts
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Sensitivity analysis
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Local sensitivity analysis
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Global sensitivity analysis
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Data assimilation
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Variational methods
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Stochastic methods
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Model reduction
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Gaussian processes
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Polynomial Chaos
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UE Temporal, spatial and extreme event analysis
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Modelling extreme temperatures, extreme river flows, earthquakes intensities, neuronal activity, map diseases, lightning strikes, forest fires, for example is a risk modelling and assessment task, which is tackled in statistics using point processes and extreme value theory.
On the one hand, point processes are a class of stochastic processes modelling random events in interaction. By event we can think of the time a neuron activates, an earthquake occurs, the time a tweet has been retweeted, etc or the location of a tree in a forest, the impact of a lightning strike, etc. The first two parts provide an introduction to stochastic models and statistical inference which could cover such applications. Main characteristics of such processes, standard models (properties, simulation) and statistical procedures to infer them will be presented.
On the other hand, taking into account extreme events such as heavy rainfalls, floods, extreme temperatures is often crucial in the statistical approach to risk modeling. In this context, the behavior of the distribution tail is then more important than the shape of the central part of the distribution. Extreme-value theory offers a wide range of tools for modeling and estimating the probability of extreme events.
UE Advanced Machine Learning: Applications to Vision, Audio and Text
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The course is split into two parts. During the first part, a wide range of machine learning algorithms will be discussed. The second part will focus on deep learning, and presentations more applied to the three data modalities and their combinations. The following is a non-exhaustive list of topics discussed:
- Computing dot products in high dimension & Page Rank
- Matrix completion/factorization (Stochastic Gradient Descent, SVD)
- Monte-carlo, MCMC methods: Metropolis-Hastings and Gibbs Sampling
- Unsupervised classification: Partitionning, Hierarchical, Kernel and Spectral clustering
- Alignment and matching algorithms (local/global, pairwise/multiple), dynamic programming, Hungarian algorithm,…
- Introduction to Deep Learning concepts, including CNN, RNN, Metric learning
- Attention models: Self-attention, Transformers
- Auditory data: Representation, sound source localisation and separation.
- Natural language data: Representation, Seq2Seq, Word2Vec, Machine Translation, Pre-training strategies, Benchmarks and evaluation
- Visual data: image and video representation, recap of traditional features, state-of-the-art neural architectures for feature extraction
- Object detection and recognition, action recognition.
- Multimodal learning: audio-visual data representation, multimedia retrieval.
- Generative Adversarial Networks: Image-image translation, conditional generation
UE Natural Language Processing & Information Retrieval
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The automatic processing of languages, whether written or spoken, has always been an essential part of artificial intelligence. This domain has encouraged the emergence of new uses thanks to the arrival in the industrial field of many technologies from research (spell-checkers, speech synthesis, speech recognition, machine translation, …). In this course, we present the most recent advances and challenges for research. We will discuss discourse analysis whether written or spoken, text clarification, automatic speech transcription and automatic translation, in particular recent advances with neural models.
Information access and retrieval is now ubiquitous in everyday life through search engines, recommendation systems, or technological and commercial surveillance, in many application domains either general or specific like health for instance. In this course, we will cover Information retrieval basics, information retrieval evaluation, models for information retrieval, medical information retrieval, and deep learning for multimedia indexing and retrieval
UE From Basic Machine Learning models to Advanced Kernel Learning
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Statistical learning is about the construction and study of systems that can automatically learn from data. With the emergence of massive datasets commonly encountered today, the need for powerful machine learning is of acute importance. Examples of successful applications include effective web search, anti-spam software, computer vision, robotics, practical speech recognition, and a deeper understanding of the human genome. This course gives an introduction to this exciting field. In the first part, we will introduce basic techniques such as logistic regression, multilayer perceptrons, nearest neighbor approaches, both from a theoretical and methodological point of views. In the second part, we will focus on more advanced techniques such as kernel methods, which is a versatile tool to represent data, in combination with (un)supervised learning techniques that are agnostic to the type of data that is learned from. The learning techniques that will be covered include regression, classification, clustering and dimension reduction. We will cover both the theoretical underpinnings of kernels, as well as a series of kernels that are important in practical applications. Finally we will touch upon topics of active research, such as large-scale kernel methods and the use of kernel methods to develop theoretical foundations of deep learning models.
UE Mathematical Foundations of Machine Learning
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Machine Learning is one of the key areas of Artificial Intelligence and it concerns the study and the development of quantitative models that enables a computer to perform tasks without being explicitly programmed to do them. Learning in this context is hence to recognize complex forms and to make intelligent decisions. Given all existing entries, the difficulty of this task lies in the fact that all possible decisions is usually very complex to enumerate. To get around that, machine learning algorithms are designed in order to gain knowledge on the problem to be addressed based on a limited set of observed data extracted from this problem. To illustrate this principle, consider the supervised learning task, where the prediction function, which infers a predicted output for a given input, is learned over a finite set of labeled training examples, where each instance of this set is a pair constituted of a vector characterizing an observation in a given vector space, and an associated desired response for that instance (also called desired output). After the training step, the function returned by the algorithm is sought to give predictions on new examples, which have not been used in the learning process, with the lowest probability of error. The underlying assumption in this case is that the examples are, in general, representative of the prediction problem on which the function will be applied. We expect that the learning algorithm produces a function that will have a good generalization performance and not the one that is able to perfectly reproduce the outputs associated to the training examples. Guarantees of learnability of this process were studied in the theory of machine learning largely initiated by Vladimir Vapnik. These guarantees are dependent on the size of the training set and the complexity of the class of functions where the algorithm searches for the prediction function. Emerging technologies, particularly those related to the development of Internet, reshaped the domain of machine learning with new learning frameworks that have been studied to better tackle the related problems. One of these frameworks concerns the problem of learning with partially labeled data, or semi-supervised learning, which development is motivated by the effort that has to be made to construct labeled training sets for some problems, while large amount of unlabeled data can be gathered easily for these problems. The inherent assumption, in this case, is that unlabeled data contain relevant information about the task that has to be solved, and that it is a natural idea to try to extract this information so as to provide the learning algorithm more evidence. From these facts were born a number of works that intended to use a small amount of labeled data simultaneously with a large amount of unlabeled data to learn a prediction function.
The intent of this course is to propose a broad introduction to the field of Machine Learning, including discussions of each of the major frameworks, supervised, unsupervised, semi-supervised and reinforcement learning.
UE Statistical learning: from parametric to nonparametric models
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course is related to mathematical and statistical methods which are very used in supervised learning.
It contains two parts.
In the first part, we will focus on parametric modeling. Starting with the classical linear regression, we will describe several families of estimators that work when considering high-dimensional data, where the classical least square estimator does not work. Model selection and model assessment will particularly be described.
In the second part, we shall focus on nonparametric methods. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for non parametric regression from independent to correlated training dataset. We shall also study some methods to avoid the overfitting in supervised learning.
This course will be followed by practical sessions with the R software.
UE Learning, Probabilities and Causality
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
Causality is at the core of our vision of the world and of the way we reason. It has long been recognized as an important concept and was already mentioned in the ancient Hindu scriptures: “Cause is the effect concealed, effect is the cause revealed”. Even Democritus famously proclaimed that he would rather discover a causal relation than be the king of presumably the wealthiest empire of his time. Nowadays, causality is seen as an ideal way to explain observed phenomena and to provide tools to reason on possible outcomes of interventions and what-if experiments, which are central to counterfactual reasoning, as ‘‘what if this patient had been given this particular treatment?’’
UE Efficient methods in optimization
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The subject of this half-semester course are more advanced methods in convex optimization. It consists of 6 lectures, 2 x 1,5 hours each, and can be seen as continuation of the course “Non-smooth methods in convex optimization”.
This course deals with:
Evaluation : A two-hours written exam (E1) in December. For those who do not pass there will be another two-hours exam (E2) in session 2 in spring.
UE Data Science Seminars and Challenge
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This course contains two parts.
Part I concerns Data challenge.
This part consists in a real problem that is given to the students for which data are readily available. The goal is to have teams of five to six students compete in solving (at least partially) the problem.
The work is spread over the Autumn semester and consists of: building a prediction model or a methodology to solve the problem based on a set of training data, blind evaluation of the model or methodology on a test bench (unseen data, withheld from the students), using an appropriate performance measure.
At the end, the teams will present their solution path in a formal presentation and a short report.
Part II concerns Data Science seminars.
This is a cycle of seminars or presentations with a common factor that is the project of the data challenge. A first seminar will settle the context and the problem for that year’s data challenge.
The other seminars will propose different industrial or academic approaches and problems that are (loosely) related to the objective of the data challenge. Presentations have a time slot of one hour and students will have to read up front some ressources to orient their questions about the subject after the seminar.
UE Computational biology
Level
Baccalaureate +5
ECTS
3 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
This interdisciplinary MSc course is designed for applicants with a biomedical, computational or mathematical background. It provides students with the necessary skills to produce effective research in bioinformatics and computational biology.
UE Quantum Information & Dynamics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Automne
The quantum formalism developed a century ago provides a very precise description of nature at small scales which entails several counter intuitive aspects: superposition of states, entanglement, intrinsic randomness of measurement process, to list a few. However, from a mathematical point of view, quantum mechanics does have a definite formulation. This allows to investigate these intriguing features rigorously and to explore these quantum traits in information theory and algorithmics in particular, as well as the challenges they present.
The goal of these lectures is to provide a mathematical description of the quantum formalism in finite dimension and to introduce the mathematical concepts and tools required for the analysis of such quantum systems and their dynamics. On the one hand, we will study the key aspects of quantum information theory. On the other hand, we will describe certain properties of quantum dynamics that need to be taken into account in the implementation of quantum algorithms and that will be applied to emblematic systems. The interaction with an external classical electromagnetic field will also be considered both from a theoretical and a numerical point of view.
UE Numerical Mechanics
Level
Baccalaureate +5
ECTS
6 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées), Grenoble INP - Ensimag (Informatique, mathématiques appliquées et télécommunications), UGA
Semester
Automne
UE Research projects
Level
Baccalaureate +5
ECTS
30 credits
Component
UFR IM2AG (informatique, mathématiques et mathématiques appliquées)
Semester
Printemps
This is the master thesis project.
Admission
Access conditions
The first year master is open to students with a degree conferring the title of bachelor in a field compatible with the fields of the master, or with a validation of studies or of prior experience.
Admission to the second year's master is selective. It is open to candidates who completed a first year master in the field.
Continuing education : You are in this situation if :
- you resume your studies after 2 years or more of interruption of studies
- or you followed a formation under the regime Formation continue during one of the 2 preceding years
- or you are an employee, job seeker, self-employed
If you do not have the diploma required to integrate the training program, you can undertake a validation of personal and professional achievements (VAPP). (in French)
Candidature / Application
Would you like to apply and register ? Then please be aware that the procedure differs depending on your diploma, on your degree, or on your place of residence for foreign students. For more details, please follow this link (in French)
And after
Sector(s)
Activity areas : R & D, mathematical engineering in industry, public research, and education
Targeted trades
The main opportunities for each standard programme are:
-Preparation for agregation : algebra, analysis, modelling, teacher and agregation
- Science in industrial and applied mathematics: researcher and teacher, researcher in applied mathematics, R&D engineer in mathematics and industrial computing, technical and commercial engineer
- Statistics and data sciences : statistical engineer, data-analyst, biostatistician, statistical programmer in industry and administration, technical commercial and statistical engineer
- ORCO : operational research engineer, logistics engineer, optimization development engineer, R&D engineer in operations research, teacher-researcher in operations research and combinatorics
- Cybersecurity : engineer in cybersecurity, security of information systems, specialized in auditing security of information systems, technical engineer in computer security, R&D engineer specialized in cybersecurity
- Fundamental mathematics : researcher and teacher-researcher in mathematics, higher education
Additional information
Several courses (MSIAM, CySec, ORCO) provide highly sought-after math/computing skills.