Master of Science in industrial and applied mathematics (MSIAM)

This course is an introduction to the main concepts of object-oriented programming, elaborated on C++. It mainly considers: Basics on classes, instances, constructors and destructors, aggregation. Memory management, pointers, references. Operator overloading. Genericity, template classes. STL (Standard Template Library) objects. Inheritance, polymorphism. The objective of this course is to present the computer sciences basics useful for applied mathematics.

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Give an overview of modelling using partial differential equations.

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.

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The aim of this course is to provide the basics mathematical tools and methods of image processing and applications.

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.

Voir la page complète

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.

Differential geometry of curves

Approximation of curves with splines, Bézier and spline curves, algorithms,…

Differential geometry of surfaces, metric and curvature properties,…

This course includes practical sessions.

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The aim of this course is to provide basic knowledge of applied probability and an introduction to mathematical statistics.

Applied probability

Estimation (parameter)

Sample comparison

Statistical tests

This course includes practical sessions.

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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

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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.

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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

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The aim of this course is to give an introduction to numerical and computing problematics of large dimension problems.

Contents:

• Introduction to database
• Introduction to big data
• Introduction to high performance computing (HPC)
• Numerical solvers for HPC

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Introduction to high performance computinb (HPC)

Numerical solvers for HPC

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Introduction to database

Introduction to big data

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January science and/or industrial project.

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Industrial and/or research internship.

The students have to do an internship (of at least 8 weeks from mid May to end of August, see the planning) in a company or in a laboratory. No report is required (except for Ensimag students that follow the double diploma, who have to give a report to ensimag).

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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

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The main objective of this course is to provide basics tools in operations research

- Contents

What is OR?
Linear Programming
Duality
Mixed Integer Programming
Dynamic programming
Constraint Programming
Complexity theory and Scheduling

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Skills

• Recognize a situation where Operations Research is relevant.
• Know the main tools of Operations Research.
• Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
• Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

• Linear Programming (modelling, solving, duality)
• Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
• Dynamic Programming
• Bonus (riddles, elsewhere on the web, OR News)

More details : https://moodle.caseine.org/course/view.php?id=42

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In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport. For each of these themes, we will see how these theoretical results relate to practical operations research problem and finally illustrate them numerically in Python.

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The aim of this course is to give mathematical grounds of security, integrity, authentication and cryptology.

Course

1. Binary encoding of information
2. Zn* group, field theory
3. Symmetric cryptography
4. Asymmetric cryptography, RSA
5. Hash, DSA
6. Lossless compression
7. Error correcting codes
8. Linear codes
9. Cyclic codes

This course include practical sessions.

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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.

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The aim of this course is to give mathematical grounds and algorithms for the modelling, animation, and synthesis of images.

Content

1. Projective rendering methods
2. Animation, cinematic methods
3. Geometrical modelling, 3D, deformation
4. Case study
5. This course include practical sessions. Implementation using OpenGL.

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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.

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Ce cours a deux objectifs principaux: 

1- Fournir un panorama assez vaste des applications de la dynamique des fluides neutres (hydrodynamique ou HD) en astrophysique. Seront ainsi abordés:

- l’équilibre des systèmes auto-gravitants, l’effondrement gravitationnel et la formation des disques circumstellaires

- les écoulements supersoniques et points critiques associés, la formation des chocs et ondes de détonation (supernovae)    

- la théorie des disques d’accrétion turbulents autour des trous noirs et des étoiles en formation

2- Fournir les hypothèses et les équations maitresses de la magnéto-hydrodynamique (ou MHD), qui est une description monofluide des plasmas. Quelques applications astrophysiques seront ensuite abordées:
 - ondes magnétiques d'Alfven et magnétosoniques    

- bouclier magnétique s’opposant à l’effondrement des nuages

Nous aborderons également, en fin de cours, ce qui est la certainement l’expérience de physique la plus longue de l’humanité, aux conséquences potentiellement majeures sur le devenir de nos sociétés, à savoir la production d’énergie électrique par fusion thermonucléaire contrôlée (ITER et autres machines à confinement magnétique).  

Au-delà de la compréhension nouvelle de certains phénomènes en astrophysique, ce cours illustre la puissance descriptive des plasmas (ici HD ou MHD). A l'issue de ce cours, des concepts avancés tels que les caractéristiques dans des écoulements hyperboliques, les méthodes perturbatives dans des régimes complexes, la turbulence ou encore les ondes dans des milieux inhomogènes auront été abordés.  

Voir la page complète

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The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Content

1. Introduction to modelling with examples.
2. Boundary problem in 1D, variational formulation, Sobolev spaces.
3. Stationary problem, elliptic equations.
4. Finite element method: algorithm, errors…
5. Evolution models, parabolic equations, splitting methods
6. Extensions and applications, FreeFEM++

This course include practical sessions.

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Course contents:

• Introduction to modelling with examples.
• Boundary problem in 1D, variational formulation, Sobolev spaces.
• Stationary problem, elliptic equations.
• Finite element method: algorithm, errors…
• Evolution models, parabolic equations, splitting methods
• Extensions and applications, FreeFEM++

This course include practical sessions.

This is a two parts course:

1. Course mutualized with Ensimag 2A 4MMMVAM (head: Emmanuel Maitre)
2. MSIAM specific course (in-depth and practical session) (head: Clément Jourdana)

A description of the course is available here

Voir la page complète

Voir la page complète

This course is an introduction to the main concepts of object-oriented programming, elaborated on C++. It mainly considers: Basics on classes, instances, constructors and destructors, aggregation. Memory management, pointers, references. Operator overloading. Genericity, template classes. STL (Standard Template Library) objects. Inheritance, polymorphism. The objective of this course is to present the computer sciences basics useful for applied mathematics.

Voir la page complète

Give an overview of modelling using partial differential equations.

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.

Voir la page complète

Voir la page complète

Voir la page complète

The aim of this course is to provide the basics mathematical tools and methods of image processing and applications.

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.

Voir la page complète

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.

Differential geometry of curves

Approximation of curves with splines, Bézier and spline curves, algorithms,…

Differential geometry of surfaces, metric and curvature properties,…

This course includes practical sessions.

Voir la page complète

The aim of this course is to provide basic knowledge of applied probability and an introduction to mathematical statistics.

Applied probability

Estimation (parameter)

Sample comparison

Statistical tests

This course includes practical sessions.

Voir la page complète

Voir la page complète

The aim of this course is to give an introduction to numerical and computing problematics of large dimension problems.

Contents:

• Introduction to database
• Introduction to big data
• Introduction to high performance computing (HPC)
• Numerical solvers for HPC

Voir la page complète

Introduction to high performance computinb (HPC)

Numerical solvers for HPC

Voir la page complète

Introduction to database

Introduction to big data

Voir la page complète

January science and/or industrial project.

Voir la page complète

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

Voir la page complète

Voir la page complète

The aim of this course is to give mathematical grounds of security, integrity, authentication and cryptology.

Course

1. Binary encoding of information
2. Zn* group, field theory
3. Symmetric cryptography
4. Asymmetric cryptography, RSA
5. Hash, DSA
6. Lossless compression
7. Error correcting codes
8. Linear codes
9. Cyclic codes

This course include practical sessions.

Voir la page complète

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.

Voir la page complète

Voir la page complète

The main objective of this course is to provide basics tools in operations research

- Contents

What is OR?
Linear Programming
Duality
Mixed Integer Programming
Dynamic programming
Constraint Programming
Complexity theory and Scheduling

Voir la page complète

Skills

• Recognize a situation where Operations Research is relevant.
• Know the main tools of Operations Research.
• Have the methodological elements to choose the solution methods and the tools the most adapted for a given practical problem.
• Know how to manipulate the software tools to solve a discrete optimization problem.

The course covers various topics:

• Linear Programming (modelling, solving, duality)
• Mixed Integer Linear Programming (modelling techniques, solving with Branch and Bound)
• Dynamic Programming
• Bonus (riddles, elsewhere on the web, OR News)

More details : https://moodle.caseine.org/course/view.php?id=42

Voir la page complète

In this part, we will investigate in more details some mathematical notions related to operations research. We will focus on three aspects: Spectral graph theory, Game theory, and Numerical Optimal transport. For each of these themes, we will see how these theoretical results relate to practical operations research problem and finally illustrate them numerically in Python.

Voir la page complète

The aim of this course is to give mathematical grounds and algorithms for the modelling, animation, and synthesis of images.

Content

1. Projective rendering methods
2. Animation, cinematic methods
3. Geometrical modelling, 3D, deformation
4. Case study
5. This course include practical sessions. Implementation using OpenGL.

Voir la page complète

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.

Voir la page complète

Voir la page complète

Voir la page complète

Ce cours a deux objectifs principaux: 

1- Fournir un panorama assez vaste des applications de la dynamique des fluides neutres (hydrodynamique ou HD) en astrophysique. Seront ainsi abordés:

- l’équilibre des systèmes auto-gravitants, l’effondrement gravitationnel et la formation des disques circumstellaires

- les écoulements supersoniques et points critiques associés, la formation des chocs et ondes de détonation (supernovae)    

- la théorie des disques d’accrétion turbulents autour des trous noirs et des étoiles en formation

2- Fournir les hypothèses et les équations maitresses de la magnéto-hydrodynamique (ou MHD), qui est une description monofluide des plasmas. Quelques applications astrophysiques seront ensuite abordées:
 - ondes magnétiques d'Alfven et magnétosoniques    

- bouclier magnétique s’opposant à l’effondrement des nuages

Nous aborderons également, en fin de cours, ce qui est la certainement l’expérience de physique la plus longue de l’humanité, aux conséquences potentiellement majeures sur le devenir de nos sociétés, à savoir la production d’énergie électrique par fusion thermonucléaire contrôlée (ITER et autres machines à confinement magnétique).  

Au-delà de la compréhension nouvelle de certains phénomènes en astrophysique, ce cours illustre la puissance descriptive des plasmas (ici HD ou MHD). A l'issue de ce cours, des concepts avancés tels que les caractéristiques dans des écoulements hyperboliques, les méthodes perturbatives dans des régimes complexes, la turbulence ou encore les ondes dans des milieux inhomogènes auront été abordés.  

Voir la page complète

Voir la page complète

Voir la page complète

Voir la page complète

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Content

1. Introduction to modelling with examples.
2. Boundary problem in 1D, variational formulation, Sobolev spaces.
3. Stationary problem, elliptic equations.
4. Finite element method: algorithm, errors…
5. Evolution models, parabolic equations, splitting methods
6. Extensions and applications, FreeFEM++

This course include practical sessions.

Voir la page complète

The aim of this course is to get deep knowledge of PDE modelling and their numerical resolution, in particular using variational methods such as the Finite Elements method.

Course contents:

• Introduction to modelling with examples.
• Boundary problem in 1D, variational formulation, Sobolev spaces.
• Stationary problem, elliptic equations.
• Finite element method: algorithm, errors…
• Evolution models, parabolic equations, splitting methods
• Extensions and applications, FreeFEM++

This course include practical sessions.

This is a two parts course:

1. Course mutualized with Ensimag 2A 4MMMVAM (head: Emmanuel Maitre)
2. MSIAM specific course (in-depth and practical session) (head: Clément Jourdana)

A description of the course is available here

Voir la page complète

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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

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This lecture proposes various industrial modeling problems and their solutions. Students are faced to an industrial problem. They are in charge of this 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.

This lecture introduces basic communication methodes in industry.

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.

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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

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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.

The purpose of this course is to discuss the main aspects related to the numerical resolution and the practical implementation of shape and topology optimization problems, and to present state-of-the-art elements of response. It focuses as well on the needed theoretical ingredients as on the related numerical considerations. More specifically, the following issues will be addressed:

- How to define a `good' notion of derivative for a ``cost'' function depending on the domain;
-  How to calculate the shape derivative of a function which depends on the domain

via the solution of a Partial Differential Equation posed on it;

 - How to devise efficient first-order algorithms (e.g. steepest-descent algorithms) based on the notion of shape derivative;
 - How to numerically represent shapes so that it is at the same time convenient to perform Finite Element computations on them,

and to deal with their evolution in the course of the optimization process.

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Cours de remise à niveau.

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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.

1. Introduction to parallelism
2. Introduction to general context of parallelism
3. Models of parallel programming
4. Description of various model of parallelism
5. Paradigm of parallelism
6. Templates of parallelism
7. Parallel architectures
8. Programming tools: Cuda

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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

1. Differentiability on normed vector spaces
2. Image restoration
3. Edge detection

Part II: Wavelets and Applications

1. From Fourier to the 1D Continuous Wavelet Transform
2. Wavelet zoom, a local characterization of functions
3. The 2D Continuous Wavelet Transform
4. The 1D and 2D Discrete Wavelet Transform
5. Linear and nonlinear approximations in wavelet bases
6. The graph Fourier and wavelets transforms

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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:

1. environmental problems: yield stress fluids (Bingham type) for granular matter, e.g. snow avalanches, mud or ice flows, erosion, landslides and volcanic lavas.

2. industrial problems: viscoelastic fluids (Oldroyd type) for plastic material processes, and metallic alloy.

3. 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.

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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

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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.

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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

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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

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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.

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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.

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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.

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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?’’

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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.

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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.

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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.

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This is the master thesis project.

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Cette UE est obligatoire pour les étudiant.e.s inscrit.e.s dans le PT MSTIC.
Elle est constituée d'une projet tutoré et d'un MOOC consacré à l'éthique de la Science. 

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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

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This lecture proposes various industrial modeling problems and their solutions. Students are faced to an industrial problem. They are in charge of this 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.

This lecture introduces basic communication methodes in industry.

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.

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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

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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.

The purpose of this course is to discuss the main aspects related to the numerical resolution and the practical implementation of shape and topology optimization problems, and to present state-of-the-art elements of response. It focuses as well on the needed theoretical ingredients as on the related numerical considerations. More specifically, the following issues will be addressed:

- How to define a `good' notion of derivative for a ``cost'' function depending on the domain;
-  How to calculate the shape derivative of a function which depends on the domain

via the solution of a Partial Differential Equation posed on it;

 - How to devise efficient first-order algorithms (e.g. steepest-descent algorithms) based on the notion of shape derivative;
 - How to numerically represent shapes so that it is at the same time convenient to perform Finite Element computations on them,

and to deal with their evolution in the course of the optimization process.

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Cours de remise à niveau.

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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.

1. Introduction to parallelism
2. Introduction to general context of parallelism
3. Models of parallel programming
4. Description of various model of parallelism
5. Paradigm of parallelism
6. Templates of parallelism
7. Parallel architectures
8. Programming tools: Cuda

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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

1. Differentiability on normed vector spaces
2. Image restoration
3. Edge detection

Part II: Wavelets and Applications

1. From Fourier to the 1D Continuous Wavelet Transform
2. Wavelet zoom, a local characterization of functions
3. The 2D Continuous Wavelet Transform
4. The 1D and 2D Discrete Wavelet Transform
5. Linear and nonlinear approximations in wavelet bases
6. The graph Fourier and wavelets transforms

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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:

1. environmental problems: yield stress fluids (Bingham type) for granular matter, e.g. snow avalanches, mud or ice flows, erosion, landslides and volcanic lavas.

2. industrial problems: viscoelastic fluids (Oldroyd type) for plastic material processes, and metallic alloy.

3. 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.

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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

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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.

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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

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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

Voir la page complète

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.

Voir la page complète

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.

Voir la page complète

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.

Voir la page complète

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?’’

Voir la page complète

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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.

Voir la page complète

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.

Voir la page complète

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.

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This is the master thesis project.

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