# 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)

## Advanced learning models

3 ECTS, C. 18h, Julien Mairal and Jakob Verbeek

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 methods as a versatile tool to represent data, and recent convolutional and recurrent neural network models for visual recognition and sequence modeling.

http://thoth.inrialpes.fr/people/mairal/teaching/2017-2018/MSIAM

Detailed description of GBX9AM16 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Bayesian statistics

3 ECTS, C. 18h, Julyan Arbel

#### Objectives

The course aims at providing an overview of Bayesian parametric and nonparametric statistics. Students will learn how to model statistical and machine learning problems from a Bayesian perspective and study theoretical properties of the models.

#### Syllabus

This course is in two parts covering fundamentals of Bayesian parametric and nonparametric inference, respectively. It focuses on the key probabilistic concepts and stochastic modelling tools at the basis of the most recent advances in the field.

##### Part 1

Foundations of Bayesian inference: exchangeability, de Finetti's representation theorem

Conjugacy in simple models (binomial, Poisson, Gaussian)

Some elements of posterior sampling, Markov chain Monte Carlo

Bayesian neural networks and their Gaussian process limit

##### Part 2

Clustering and Dirichlet process, random partitions

Models beyond the Dirichlet process, random measures, Indian buffet process

Some elements of Bayesian asymptotics

#### Bibliography

Hoff, P. D. (2009). A first course in Bayesian statistical methods. Springer Science & Business Media.

Neal, R. M. (2012). Bayesian learning for neural networks (Vol. 118). Springer Science & Business Media.

Hjort, N. L., Holmes, C., Müller, P., & Walker, S. G. (2010). Bayesian nonparametrics. Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge Univ. Press.

Orbanz, P. (2012). Lecture Notes on Bayesian Nonparametrics. Available at: http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf

Kleijn, B., van der Vaart, A., & van Zanten, H. (2012). Lectures on Nonparametric Bayesian Statistics. Available at: https://staff.fnwi.uva.nl/b.j.k.kleijn/NPBayes-LecNotes-2015.pdf

http://www.julyanarbel.com/bayesian-statistics-course

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

## Category learning and object recognition

3 ECTS, C. 18h, Jakob Verbeek

In this course we present recent state-of-the-art methods for visual object category representation and recognition, and the techniques that underpin these methods. Methods will include so called “bag of features” approaches, Fisher vectors, and convolutional neural networks for tasks such as instance-level image retrieval, image classification, object localization, semantic segmentation, image
caption generation and action recognition in videos.
On the machine learning side we consider clustering methods (k-means, mixutre of Gaussians), classification techniques (SVM, logistic discriminant), and kernels to obtain non-linear classifiers, as well as the principles underlying neural networks (multi-layer perceptron, back-propagation, convolutional networks, recurrent networks).

Detailed description of GBX9MO31 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Computational biology

3 ECTS, C. 18h, Olivier François and Michaël Blum

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)

## Convex and distributed optimization

3 ECTS, 12h courses, 21h practical work, Franck Iutzeler, Jerome Malick and T. Ropars

Courses: (4 parts of 3h each + 3 hours practical work)

Introduction to convex optimization: concepts in convex analysis (duality, proximal operators), how to identify potential difficulties in optimization problems. Illustrations in supervised learning (classification and regression problems) and in operation research (decomposition methods).

Algorithms in convex optimization (gradient, proximal gradient, conditional gradient, ADMM)

Introduction to distributed computation (architectures for computation, map-reduce scheme, MPI, Spark) + 3h practical work

Distributed optimisation algorithms, stochastic algorithms, asynchronous methods.

Practical work (2 parts of 6h each)

application to a recommendation system

sparse logistic regression in high dimension

Detailed description of GBX9MO15 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Data Challenges

3 ECTS, Project, Jean-Baptiste Durand, Ronald Phlypo and Olivier Michel

Face up challenging real-world problems in machine learning, be involved in multidisciplinary teams of data scientists, computer scientists, mathematicians and expert students in signal processing, and contribute to leading your team to the top rank!

Different teams with M2 students issued from either MSIAM Data Science, MoSIG Data Science and SIGMA work on a same challenge on either complex, structured or big data, and maybe a combination of all three. Try and compare different approaches, take benefit from the computational power of clusters and from advice of your supervisors.

The data challenges stretch on several months, include some tutored sessions, if needed mini-courses, and of course your regular involvement over that period of time.

Organization of the data challenge in 2018-2019

Detailed description of GBX9AM20 ; 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, Jean-Baptiste Durand, Ronald Phlypo and Olivier Michel

Our master programs now include a series of 6 or 7 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.

At the end of the seminar series, some oral exam is organized. One of the topic presented during the seminars is randomly assigned to each student a few days in advance. The oral exam consists in a 25 min summarized presentation of the scientific issues that were addressed, and a 15 min session of discussion and questions. A second different topic is chosen by the student, and he.she must write a report on that topic, based on the seminar and associated articles.

The seminars will be on Thursdays around 3:30PM (no sooner).

Follow the announcements on https://data-institute.univ-grenoble-alpes.fr/education/data-science-seminar-series/ (regularly updated)

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

## Distributed Systems

3 ECTS, 18h, Vivien Quéma.

Review of core principles of distributed systems

Characteristics and design issues of distributed systems, briefly revisiting the basic notions on network support, naming and binding.

Main concepts and terminology of fault tolerance, including replicated servers for high-availability.

Peer-to-peer Distributed Systems

Both structured and unstructured P2P architectures and designs.

Detailed description of GBX9MO39 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## 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, Jean-Baptiste Durand

This lecture introduces fundamental concepts and associated numerical methods in model-based clustering, classification and models with latent structure. These approaches are particularly relevant to model random vectors, sequences or graphs, to account for data heterogeneity, and to present general principles in statistical modelling. The following topics are addressed:

Principles of probabilistic data mining and generative models; models with latent variables

Probabilistic graphical models

Mixture models and clustering

PCA and probabilistic PCA

Generative models for series and graphs : hidden Markov models

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

## 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. 9h, L. 9h, 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.
Through different lab assignments, the students will apply the concepts of efficient parallel programming using distributed and shared memory programming language (OpenMP, MPI).
In the final project, the students will have the possibility to parallelize one of their own numerical application developed in a previous course.

Detailed description of GBX9MO16 ; 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)

## Inverse methods and data assimilation

3 ECTS, C. 18h, Labs 12h. Elise Arnaud

Forecasting the weather, identifying the optimal shape of an aircraft wing, medical
imagery, and more generally determining the value of some unknown parameters in a
system given a mathematical model and/or observations, is an inverse problem. Methods
for addressing such problems are described in this course. These methods are based
either on optimal control theory or on statistical estimation theory.

Detailed descriptionf of GBX9AMO6 ; MSIAM Course list ; Semester 3 (MSIAM tracks)

## Machine learning fundamentals

3 ECTS 18h, Massih-Rezah Amini

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)

## Medical Imaging: tomography and 3D reconstruction from 2D projections

3 ECTS, C. 18h, Laurent Desbat and Rolf Clackdoyle

CT Scanners and nuclear imaging (SPECT and PET) have greatly improved medical
diagnoses and surgical planning. Mathematics is necessary for these medical imaging
systems to deliver images. We present mathematical problems arising from these
medical imaging systems. We show how to reconstruct images from projections of the
attenuation function in radiology or respectively of the activity in nuclear imaging. We
present recent advances in 2D and 3D reconstruction problems.

Detailed description of GBX9AM07 ; MSIAM Course list ; 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)

## Optimal Transport, levelset: applications to image

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)

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)

## 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, Valérie Perrier

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, …).