# M1 Course offer

### Object-oriented & software design

##### Objective

The objective of this course is to present the computer sciences basics useful for applied mathematics.

##### Content

Compilation (const, inline, loops, Gnu Make …)

C++: genericity (template), code reuse (STL), efficient programming

Objects and hierarchical memory, notions of cache and locality (e.g., BLAS)

Basics of algorithmics

Complexity

Error propagation, floating point computing

This course relies on practical sessions.

##### Grading

1/2 practical

1/2 final written exam

##### Prerequisite

Good knowledge of C programming (including low-level concepts such as pointers and memory allocation)

Course content
(head: Laurence Pierre)

### Applied probability and Statistics

6 ECTS, CM 24h, TP 24h

Mutualized with M1 SSD Applied probability and Statistics

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

Contents:

Applied probability

Estimation (parameter)

Sample comparison

Statistical tests

This course includes practical sessions.

Grading:

See the contents from the associated course at UPMF.

Course content for probability

Course content for statistics

### Partial differential equations and numerical methods

6 ECTS, CM 16.5h, TD 16.5h, TP 16.5h

##### Objectives

Give an overview of modelling using partial differential equations.

##### Content

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.

##### Course Organization

##### Grading

1/2 practical

1/2 final written exam

##### Prerequisite

Basic notions of real analysis, including Taylor formula, functions of several real variables and partial derivatives
Methods for solving first order ordinary differential equations (linear case, variation of constants method, separable ODEs…)
Basic notions on Fourier series and Fourier transform

Course content (part 1)
Course content (part 2)

### Signal and image processing

6 ECTS, CTD 36h, TP 18h

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

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.

Grading:

1/2 practical

1/2 final written exam

(head: Cécile Amblard)

### Geometric modelling

6 ECTS, CTD 36h, TP 18h

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

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.

##### Grading

1/2 practical

1/2 final written exam

##### Prerequisite

Elementary notions of linear algebra and analysis.

(head: Boris Thibert)

### Français langue étrangère

### English

3 ECTS

English for french-speaking students.

(head: Virginia Gardner)

6 ECTS, CTD 18h, TP 36h

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

##### Content

This course relies on practical sessions.

(head: Christophe Picard)

##### Prerequisites

### Modeling activity 1: Project

### Modeling activity 2: Internship

3 ECTS

Industrial and/or research internship.

### Numerical Optimization: mathematical background and case studies

6 ECTS, CTD 36h, TP 18h

This program combines case studies coming from real life problems or models and lectures providing the mathematical and numerical backgrounds.

##### Content

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: penalisation, SQP methods

Applications and case studies

This course includes practical sessions.

Lab

Main results

(head: Laurent Desbat)

##### Prerequisite

Basic algebra (linear spaces, matrix computation)
Basic calculus (Norm, Banach spaces, Hilbert spaces, basic differential calculus)
The students should be able to compute the gradient and the Hessian of real functions on IR^n and also differentials of simple functions such as quadratic forms.

Présentation

### Variational methods applied to modelling

##### Objectives

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

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.

A description of the course is available here

##### Organization

3ECTS = CM 16.5h + TD 16.5h - Joined with Ensimag 2A

4MMMVAM (head: Emmanuel Maitre)

3ECTS = CTD 3h + TP 18h - MSIAM specific course (in-depth and practical session) (head: Clément Jourdana)

##### Prerequisites

notions of distribution theory, linear algebra, integral calculus, some notions of programming in some high level language, basic numerical analysis, as numerical integration of differential equations, basic notions on Hilbert spaces, usual partial differential operators (gradient, divergence, laplacian…)

### 3D Graphics

##### Objectives

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

##### Content

Projective rendering methods

Animation, cinematic methods

Geometrical modelling, 3D, deformation

Case study

This course include practical sessions. Implementation using OpenGL.

##### Organization

3ECTS = Lecture 16.5h + Lab 16.5h - Course joined with M1 MoSIG

3ECTS = Lecture 19.5h + Lab 1.5h - MSIAM specific course (in-depth and practical session)

A description of the course is available here

##### Prerequisites

Programming skills using a high level language

### Computer Algebra and Cryptology

##### Objectives

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

##### Course

This course include practical sessions.

##### Organization

3ECTS = TD 19h + TP 15h - Course joined with Pure Mathematics M1 (head: François Dahmani)

3ECTS = TD + TP - MSIAM specific course (in-depth and practical session) (head: Pierre Karpman)

A description of the course is available here

### Statistical analysis and document mining

##### Objectives

The aim of this course is to present advanced statistics and linear modelling, variance analysis and provide practical implementation

##### Content

Principal components analysis (PCA)

Classification (Linear Discr. Analysis)

Data mining (text mining)

Linear regression

Estimation and test of regression parameters

ANOVA

ANCOVA

Practical implementation

This course include practical sessions.

##### Organization

3ECTS = Lecture 13h + Practical 5h + Lab 15h - Course Joined with Ensimag 2A

4MMFDASM (head: Jean-Baptiste Durand)

3ECTS = Lecture 14h + Lab 6h - MSIAM specific course (in-depth and practical session) (head: Stéphane Girard)

A short description of the course content can be found here

##### Prerequisites

Elementary notions in probability theory (probability distribution, joint probability density function for random vectors, conditional distribution, expectation, variance, covariance, Gaussian distribution)

Elementary notions in mathematical statistics (estimator, confidence interval, statistical tests). As a bonus: simple linear regression.

Notions in linear algebra (matrix reductions).
As a bonus: elementary notions in Rstudio and the R software.

### Introduction to Operations Research

##### Course objectives

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

##### Content

##### Organization

3ECTS = CM 18h + TD 18h - Course joined with M1 MoSIG (Head: Nadia Brauner)

3ECTS = CTD 14h + TP 6h - MSIAM specific course (in-depth and practical session) (Head: Franck Iutzeler)

##### Prerequisites

Classical algorithms (sort, divide and conquer)

Algorithms complexity calculation

Programming: basic notions (variables, fonctions, if, for, while, tables)

Language Python or Java

Basic notions on graphs (basic definitions, graph search, trees, shortest paths)

Basic notions on linear algebra and matrix analysis (matrix multiplication, invertible matrix definition)

## ECTS total

30 ECTS per semester:

1st semester compulsory courses 27 ECTS

1st semester language course 3 ECTS

2nd semester compulsory courses 18 ECTS

2nd semester with choice 12 ECTS