# MSIAM course list

Students must choose courses in the list below, which covers a ** very wide range of mathematical topics**.
Faculty members will be pleased to advise students on how to construct their curriculum.
Meetings are scheduled at the start of the semester to inform them on the scope of the courses.

The list below is likely to change slightly.

Labs usually means numerical experiments with computers

## Refresher courses (0 ECTS)

### Introduction to matrix numerical analysis and numerical optimisation

## Advanced Algorithms for Machine Learning and Data Mining

3 ECTS 18h, Eric Gaussier and Ahlame Douzal

A prior algorithms (Frequent item sets) & Page Rank

Monte-carlo, MCMC methods: Metropolis-Hastings and Gibbs Sampling

Matrix Factorization (Stochastic Gradient Descent, SVD)

Generalized kmeans and its variants (Bach, Online, large scale), Kernel clustering (Support Vector Clustering), Spectral clustering

Classification and Regression Trees, Support Vector regression

Alignment and matching algorithms (local/global, pairwise/multiple), dynamic programming, Hungarian algorithm,…

Detailed description of GBX9MO22 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Advanced Imaging

3 ECTS, C. 18h, Sylvain Meignen

In this course, we will first focus on linear methods for image denoising. In this regard,
we will investigate some properties of the heat equation and of the Wiener filter. We will
then introduce nonlinear partial equations such as the PeronaMalick model for noise
removal, and some other similar models. A last part of the course will be devoted to edge
detection for which we will consider the Canny approach and, more precisely, we will deal
in details with active contours and level sets methods.

Detailed description of GBX9AM05 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## An introduction to shape and topology optimization

3 ECTS, C. 18h, Eric Bonnetier and Charles Dapogny

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:

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.

Prerequisites: Only a basic knowledge of functional analysis and scientific computing will be assumed: differential calculus, Finite Element method, etc.

3 ECTS, C. 18h, Dider Bresch

Granular flows are at the heart of phenomena such as erosion, landslides and vulcanology. The mathematical study of these complex events 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.

The main idea of this lecture is to motivate by examples interdisciplinary collaborations needed to deal with complex situations.

## Computational biology

3 ECTS, C. 18h, Olivier François and Clovis Galliez

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.

The objective is to provide a short introduction on bioinformatics modelling and advanced tools for the analysis of sequence data. The first part of the course focuses on application in molecular biology and evolution, including hierarchical clustering and the analysis of phylogenetic and population genetic data.

The second part of the course focuses on machine learning for biological data, and includes change point detection in sequences and unsupervised clustering of massive genetic data. The course is evaluated with two lab-works, one for each part of the course.

No specific prerequisites.

Detailed description of GBX9AM18; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Data management in large-scale distributed systems

3 ECTS, 18h, Thomas Ropars.

Target skills :
Data management and knowledge extraction have become the core activities of most organizations. The increasing speed at which systems and users generate data has led to many interesting challenges, both in the industry and in the research community.

The data management infrastructure is growing fast, leading to the creation of large data centers and federations of data centers. These can no longer be handled exclusively with classic DBMS. It requires a variety of flexible data models (relational, NoSQL…), consistency semantics and algorithms issued by the database and distributed system communities. In addition, large-scale systems are more prone to failures, and should implement appropriate fault tolerance mechanisms.

The dissemination of an increasing amount of sensors and devices in our environment highly contribute to the “Big Data” and the development of ubiquitous information systems. Data is processed in continuous streams providing information related of users context, such as their movement patterns and their surroundings. This data can be used to improve the context awareness of mobile applications and directly target the needs of the users without requiring an explicit query.

Combining large amounts of data from different sources offers many opportunities in the domains of data mining and knowledge discovery. Heterogeneous data, once reconciled, can be used to produce new information to adapt to the behavior of users and their context, thus generating a richer and more diverse experience. As more data becomes available, innovative data analysis algorithms are conceived to provide new services, focusing on two key aspects: accuracy and scalability.

Program summary :
In this course, we will study the fundamentals and research trends of distributed data management, including distributed query evaluation, consistency models and data integration. We will give an overview of large-scale data management systems, peer-to-peer approches, MapReduce frameworks and NoSQL systems. Ubiquitous data management and crowdsourcing will also be discussed.

Detailed description of GBX9MO08 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Data science seminar

3 ECTS, Seminars, Pierre Etoré, Ronald Phlypo and Hacheme Ayasso

Our master programs now include a series of 5 seminars given by active researchers in the field of data processing methods and analysis.

These seminars are intended to give students some insights on modern problems and solutions developed in a data science framework, with applications in a variety of fields.

In order to make these seminars a most valuable experience for all students, a scientific paper dealing with the topic of the seminar will be selected by the speaker and dispatched to all students about 2 weeks before the seminar. Students are expected to read and study this paper, and to prepare questions, before attending the seminar. Presence at the seminars is compulsory for master students.

Oldies: announcements of 2019-2020 on https://data-institute.univ-grenoble-alpes.fr/education/data-science-seminar-series/ (regularly updated)

In addition please follow information about this course on the Chamillo page: https://chamilo.grenoble-inp.fr/courses/PHELMAWPMTSSP7/index.php?id_session=0

This module is common with the M2 programmes MSIAM Data Science, MoSIG Data Science and SIGMA.

## Efficient methods in optimization

3 ECTS, C. 18h, Roland Hildebrand

Theoretical foundations of convex optimization.

This course deals with:

Topic 1: convex analysis

Topic 2: convex programming

Basic notions: vector space, affine space, metric, topology, symmetry groups, linear and affine hulls, interior and closure, boundary, relative interior

Convex sets: definition, invariance properties, polyhedral sets and polytopes, simplices, convex hull, inner and outer description, algebraic properties, separation, supporting hyperplanes, extreme and exposed points, recession cone, Carathéodory number, convex cones, conic hull

Convex functions: level sets, support functions, sub-gradients, quasi-convex functions, self-concordant functions

Duality: dual vector space, conic duality, polar set, Legendre transform

Optimization problems: classification, convex programs, constraints, objective, feasibility, optimality, boundedness, duality

Linear programming: Farkas lemma, alternative, duality, simplex method

Algorithms: 1-dimensional minimization, Ellipsoid method, gradient descent methods, 2nd order methods

Conic programming: barriers, Hessian metric, duality, interior-point methods, universal barriers, homogeneous cones, symmetric cones, semi-definite programming

Relaxations: rank 1 relaxations for quadratically constrained quadratic programs, Nesterovs π/2 theorem, S-lemma, Dines theorem

Polynomial optimization: matrix-valued polynomials in one variable, Toeplitz and Hankel matrices, moments, SOS relaxations

The course is composed of 18 hours lectures.

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.

Detailed description of GBX9AM25 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Fundamentals of probabilistic data mining

3 ECTS, C. 13.5h, L. 4.5h, Xavier Alameda-Pineda

**Content:** This courses introduces probabilistic models with latent variables, and the associated algorithms to estimate the parameters and perform inference over the latent variables. Such models are used for unsupervised tasks such as clustering and source modeling, as well as for supervised tasks such as classification and regression. You will discover the basic probabilistic models as well as more advanced techniques.

The following topics are addressed:

Principles of probabilistic data mining and generative models

Latent variables and probabilistic graphical models

Mixture models

The linear-Gaussian model and probabilistic PCA

Markov models for time series with continuous and discrete latent variables

Variational inference and variational auto-encoders

At the end of the course, the student will have basic knowledge in the most common probabilistic models with latent variables. Therefore, the student will be able to perform model-based clustering, analysis and segmentation of time-series with hidden Markov models, build a graphical model associated with a given distribution, represent numerical multivariate data with missing coordinates into planes and work with state-of-the-art non- linear regression models based on variational autoencoders.

**Prerequisite:** Fundamental principles in probability theory (conditioning) and statistics (maximum likelihood estimator and its usual asymptotic properties).

**Evaluation:** The first session combines a written exam (E1) and the reports of the three practical sessions (P). The final mark of the first session is obtained as (E1+P)/2. The second session consists of only a written exam (E2) which constitutes the final grade.

Detailed description of GBX9MO17 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Geophysical imaging

3 ECTS, C. 18h, Ludovic metivier

Understanding Earth's interior mechanisms, assessing seismic hazard due to earthquakes and volcanoes, securing our access to hydrocarbon resources and monitoring CO2 storage sites, all represent crucial issues for modern societies. They all share the same need for a deep and accurate knowledge of the Earth interior, particularly the crust. We present in this course a seismic imaging method yielding unprecedented high resolution information on the subsurface structure. This seismic imaging method is formulates as an optimization problem, constrained by partial differential equations representing seismic wave propagation. We study in this course all of the constitutive elements of this method, insisting on mathematical and numerical aspects of it: wave modeling in heterogeneous media, large scale optimization methods, implementation on high performance computing parallel architectures.

## GPU Computing

3 ECTS, C 18h, Christophe Picard

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.

Syllabus:

Introduction to parallelism

Introduction to general context of parallelism

Models of parallel programming

Description of various model of parallelism

Paradigm of parallelism

Templates of parallelism

Parallel architectures

Programming tools: Cuda

Prerequisite:
C or C++, Compiling, Data structures, Architecture, Concurrency

Detailed description of GBX9AM26 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

3 ECTS, C. 18h. Georges Quenot, Philippe Mulhem and Jean-Pierre Chevallet

This course addresses advanced aspects of information access and retrieval, focusing on several points: models (probabilistic, vector-space and logical), multimedia indexing, web information retrieval, and their links with machine learning. These last parts provide opportunities to present the processing of large amount of partially structured data. Each part is illustrated on examples associated with different applications.

Course contents:

Part I. Foundations of Information Retrieval

Course 1: Information retrieval basics.
Course 2: Classical models for information retrieval.
Course 3: Natural language processing for information retrieval.
Course 4: Theoretical models for information retrieval.

Part II: Web and social networks

Course 5: Web information retrieval and evaluation.
Course 6: Social networks and information retrieval.
Course 7: Personalized and mobile information retrieval.
Course 8: Recommender systems.

Part III: Multimedia indexing and retrieval

Course 9: Visual content representation and retrieval.
Course 10: Classical machine Learning for multimedia indexing.
Course 11: Deep learning for information retrieval.
Course 12: Deep learning for multimedia indexing and retrieval.

Detailed descriptionf of GBX9MO23 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

3 ECTS, C. 18h. Renaud Blanch.

Interactive Information Visualization (InfoVis)

InfoVis is the study of interactive graphical representations of abstract data (e.g. graphs linking people in social networks, series of stock options values evolving over time).

Graphical representations are a powerfull way to leverage the human perceptual capabilities to allow the user to explore and make sense of abstract data, and also to expose findings and convey ideas.

But to be efficient, a visualization has to be designed using knowledge about the human visual perception, the characteristics of the data, the kind of task that will be performed on those data.

The aim of this course is to provide the keys, both theoretical and practical, to build usable and useful interactive visualizations.

Program summary:

foundations: human visual perception, graphical variables, data types, the visualization pipeline.

linked data: tree and graph visualization

tabular data: time series and spatial data visualization

dealing with large data: aggregation, multiple views, interaction

validating visualization: visualization tasks, evaluation

Detailed descriptionf of GBX9MO37 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Introduction to extreme-value analysis

3 ECTS, C. 18h, Stéphane Girard

Taking into account extreme events (heavy rainfalls, floods, etc.) 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.
In particular, the following points will be addressed in the course:

1) Asymptotic behavior of the largest value of a sample. Extreme-value Distribution (EVD). Maximum domains of attraction (Fréchet, Weibull and Gumbel).
Asymptotic behavior of excesses over a threshold. Generalized Pareto Distribution (GPD). Regularly varying functions.

2) Estimation of the parameters of the EVD and GPD. Hill estimator. Application to the estimation of extreme quantiles.
Illustration on simulated and real data.

Prerequisites: knowledge of statistics and probability will be assumed.

Evaluation: will be a written exam.

## Kernel methods for machine learning

3 ECTS, C. 18h, Julien Mairal

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, with a strong focus on kernels as 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.

## Level set methods and optimization algorithms with applications in imaging

3 ECTS, C. 18h, Emmanuel Maître and Charles Dapogny

This lecture will link levelset modeling of biomechanical systems (e.g. immersed elastic
membranes mechanics) with optimal transportation theory. Interpolation algorithms
based on physical knowledge of images content will be studied. Theoretical as well as
practical implementation aspects will be considered.

Detailed description of GBX9AM04 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Machine Learning for Computer Vision and Audio Processing

3 ECTS, C. 18h, Karteek Alahari and Xavier Alameda-Pineda

**Content:** In this course we present recent state-of-the-art methods for machine learning applied to visual and audio data processing tasks. We will discuss the problems related to each of these two modalities, as well as those related to fusing the two modalities. We will present classical methods as well as more advanced techniques based on deep learning. The tasks addressed will be, for instance: semantic segmentation, object detection, image classification, speech enhancement or sound source separation. We will address the following topics:

* Audio and visual features
* Bag-of-features and graph-cut
* Convolutional neural networks
* Recurrent neural networks
* Generative adversarial networks
* Deep learning applications

At the end of the course, the student will have basic knowledge in the most common state-of- the-art methods to process audio, visual and audio-visual data. The student will also learn advanced techniques to process this kind of data, and in particular the basis of deep learning methods.

**Prerequisite:** Basics on linear algebra and calculus are required to understand this course. It is interesting to take the “Machine Learning Fundamentals” course in parallel. The course on “Fundamentals of Probabilistic Data Mining” can provide a complementary perspective in some of the topics discussed here.

**Evaluation:** The first session combines a written exam (E1), the answers to the quizs (Q) and the report of the practical session (P). The final mark of the first session is obtained as 0.4*E1+0.4*Q+0.2*P. The second session consists of only a written exam (E2) which constitutes the final grade.

## Machine learning fundamentals

3 ECTS 18h, Massih-Rezah Amini and Emilie Devijver

Consistency of the Empirical Risk Minimization

Uniform Generalization Bounds and Structural Risk Minimization

Unconstrained Convex Optimization

Binary Classification algorithms (Perceptron, Adaboost, Logistic Regression, SVM) and their link with the ERM and the SRM principles

Multiclass classification

Application and experimentations

Detailed descriptionf of GBX9AM19 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Model exploration for approximation of complex, high-dimensional problems

3 ECTS, 18h, Clémentine Prieur and Olivier Zahm.

Many industrial applications invole expensive computational codes which can take weeks or months to run. It is typical for weather prediction, in aerospace sector or in the civil engineering field. There is here an important (economic) challenge to reduce the computational cost by constructing a surrogate for the input-to-output relationship.
Since only a few number of model runs is affordable, dedicated tools have been developed to exploit this type of “not-so-big” data sets. This lecture focuses on some of the most recent advances in that direction.

Prerequisites:
Basic knowledge in probability and statistics

Target skills:
The goal of this lecture is to address the difficult problem of approximating high-dimensional functions, meaning functions of a large number of parameters. The first part of the lecture is devoted to interpolation techniques via polynomial functions or via Gaussian processes. In the second part, we present two methods for reducing the dimension of the input parameters space, namely the Sliced Inverse Regression and the Ridge Function Recovery.

References
Springer Handbook on UQ, R. Ghanem, D. Higdon and H. Owhadi (Eds)

Detailed description of GBX9AM23 ; MSIAM Course list ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Model selection for large-scale learning

3 ECTS, C. 18h, Emilie Devijver

When estimating parameters in a statistical model, sharp calibration is important to get optimal performances. In this course, we will focus on the selection of estimators with respect to the data. Particularly, we will consider calibration of parameters (e.g., regularization parameter for minimization of regularized empirical risk, like Lasso or Ridge estimators) and model selection (where each estimator minimizes the empirical risk on a specified model, as mixture models with several number of clusters).

We will focus on the penalized empirical risk, where the penalty may be deterministic (as BIC or ICL) or estimated with data (as the slope heuristic).

Prerequisites:

Basic knowledges in probability and statistics

Target skills: Learn

When model selection is needed.

What can be proved theoretically for existing methods.

How those results can help in practice to choose a criterion for some specific statistical problem

How the theory can serve to define new procedures of selection.

References

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning. Data Mining, Inference, and Prediction

P. Buhlmann and S. van de Geer, Statistics for High-Dimensional Data. Methods, Theory and Applications

P. Massart, Concentration Inequalities and Model Selection

Detailed description of GBX9AM24 ; MSIAM Course list ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Modelling Seminar and Projects

** ***6* ECTS, Tut. 36h, Labs 36h. Emmanuel Maître et Cécile Lalande

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.

See also: http://chamilo.grenoble-inp.fr/main/document/document.php?cidReq=ENSIMAGWMM9AM10 (intranet: for registered students only).

MSIAM Course list ; Semester 3 (MSIAM tracks)

## Numerical optimal transport and geometry

3 ECTS, C. 18h, Boris Thibert

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 solve the following very applied problem: what is the cheapest way of sending a pile of sand to a hole, knowing the cost of transportation of each sand grain of the pile to a possible target location ? This very applied 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.

The goal of this course is to introduce basics on the optimal transport theory and to present the recent algorithms that have been shown to be very efficient. We will first focus on the discrete setting that corresponds to the transportation between discrete measures, with the entropic relaxation and the Sinkorn algorithm. We will then study the semi-discrete setting that corresponds to transporting a continuous measure to a discrete one. This has connections with computational geometry and can be solved efficiently with Newton algorithms. We will also present applications in different fields such as image processing and geometric optics.

Prerequisite: not particularly.

Evaluation: written test / presentation of a research paper.

Detailed description of GBX9AM25 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

3 ECTS, C. 9h, Labs 30h, Mourad Ismail

This lecture presents various useful applications, libraries and methods for software engineering related to applied mathematics. These include:

C++ project management, development and profiling (cmake, subversion, qtcreator, gdb, gprof, valgrind)

Linear algebra (Eigen)

User interface (Qt)

Data processing (XML)

Prototyping and interfacing using Python

Detailed descriptionf of GBX9AM27 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Statistical methods for forecasting

3 ECTS, C. 18h, Sana Louhichi

This course is related to mathematical and statistical methods for forecasting in supervised learning. We will present several tools and ingredients to predict the future value of a variable. We shall focus on methods for regression and methods for classification for inde- pendent or correlated training dataset. This course will be followed by four practical sessions with the R software.

Course outline:
Introduction. Linear methods for regression. Non linear methods for regression. Supervised classification.

Keys words and phrases:
Parametric regression, Lasso, Ridge, Nonparametric trend estimation, Kernel nonparametric models, Smoothing parameter selection, Average squared error, Mean average squared error, Mallows criterion, Cross validation, Generalized cross validation, Dependent random variables, Martingale difference sequences, Stochastic Volatility, Moment inequalities, Maximal inequali- ties, Supervised classification.

Evaluation: (1/2) project + (1/2) written exam.

## Stochastic Calculus and Applications to Finance

3 ECTS, C. 18h, Pierre Etoré

This MSc course aims at presenting the fundamental concepts of Stochastic Calculus, and the way these concepts have been used in order to build models for applications to finance. Stochastic calculus is a theory that uses Brownian motion and Itô’s integral as basic building blocks, and Itô's formula as a multipurpose tool, in order to describe and manipulate a rather large variety of continuous time Stochastic processes , called « continuous semimartingales » (Stochastic calculus for processes with jumps is out of the scope of this course). The theory of Stochastic calculus is largely due to the seminal work by K. Itô, that goes back to the 1940s and 1950s. This work has been rediscovered by economists (among them Myron Scholes) in the 1970s, giving rise to the famous Black-Scholes model. Since the late 1980s the link between Stochastic calculus and economics has been more and more formalized, giving rise to the fleld of « Mathematical Finance ».

This course requires knowledge of probability and integration theory. Some previous knowledge of Stochastic processes is welcomed. No previous knowledge of Brownian motion or Stochastic Calculus is required. The content is planned to be:

Continuous time stochastic processes, Brownian motion (definition and properties)

Continuous time martingales

Itô’s integral

Itô’s formula, Theorem of Lévy, Theorem of Girsanov

Black-Scholes model; notion of pricing and hedging

Princing and hedging formulas, illustration of the link between Stochastic Differential Equations and Partial Differential Equations inside Black-Scholes type models.

Detailed description of GBX9AM22 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Wavelets and applications

3 ECTS, C. 18h, Kevin Posilano

Wavelets are basis functions widely used in a large variety of fields: signal and image
processing, numerical schemes for partial differential equations, scientific visualization.
This course will present the construction and practical use of the wavelet transform, and
their applications to image processing : Continuous wavelet transform, Fast Wavelet
Transform (FWT), compression (JPEG2000 format), denoising, inverse problems.
The theory will be illustrated by several applications in medical imaging (segmentation,
local tomography, …).