# IAM Lectures

## (T) Efficient methods in optimization

3ECTS, Anatoli Juditsky (anatoli.iouditski@imag.fr, http://www-ljk.imag.fr/membres/Anatoli.Iouditski/cours/convex/cours.pdf)

Optimization problems arise naturally in many application fields. Whatever people do, at
some point they get a craving for organizing things in a best possible way. This intention,
converted in a mathematical form, appears to be an optimization problem of certain type
(think of, say, Optimal Diet Problem). Unfortunately, the next step, consisting of finding
a solution to the mathematical model is less trivial. At the first glance, everything looks
very simple: many commercial optimization packages are easily available and any user
can get a ``solution'' to his model just by clicking on an icon at the desktop of his PC.
One of the goals of this course is to show that, despite to their attraction, the general
optimization problems very often break the expectations of a naive user. In order to apply
these formulations successfully, it is necessary to be aware of some theory, which tells us
what we can and what we cannot do with optimization problems. The elements of this
theory can be found in each lecture of the course. It describes some of most efficient
optimization techniques available today.

## (T) Dynamical Systems, bifurcations and applications

6ECTS, Antoine Girard

This course introduces fundamental notions for the study of nonlinear dynamical
systems with an emphasis on instabilities and bifurcation phenomena. We will present
general methods for the local analysis of nonlinear models (LyapunovSchmidt reduction,
center manifold, normal forms). We will also consider hybrid dynamical systems that
result from the interaction of discrete and continuous processes.

## (N) Coupling methods

3 ECTS, E. Blayo

This course presents numerical methods for solving systems of PDEs in the context of
coupled or multiscale phenomena. These tools are based mostly on Schwarz methods for
domain decomposition, and on variational techniques.

## (N) Inverse methods and data assimilation

3 ECTS, M. Nodet and E. Blayo

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.

## (N) Numerical methods for hyperbolic equations

3 ECTS, L. Debreu and G.H. Cottet

This course will cover the design and numerical analysis of finitevolume methods for
conservation laws. It will first cover the essential properties of the underlying
mathematical models (conservation, entropy conditions, maximum principle…). It will
then describe the classical schemes (upwind, LaxWendroff…) and the techniques to
derive TVD and high order methods. The properties of the methods will be illustrated in
applications in fluids mechanics and geophysics.

## (N) Optimal Transport, levelset: applications to biophysics

3 ECTS, E.Maître

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.

## (N) Mathematical methods for wave propagation: application to inverse problems and medical imaging

3 ECTS, E. Bonnetier and F. Triki.

This course studies the propagation of electromagnetic waves, described in harmonic
regime by the Maxwell equations, and, in some particular case, by the Helmholtz
equation. We first present mathematical tools for such equations, then inverse problems
like impedance tomography and magnetic resonance imaging.

## (I) Modélisation surfacique (36h – S. Hahmann et F. Hétroy)

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

3ECTS, L. Desbat

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.

## (I) Wavelets and applications

3ECTS, 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, …).

## (I) Advanced Imaging

3ECTS, 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.

This presentation covers 2D tomography including the reconstruction of Region Of
Interest from noncomplete data (very short scan trajectories, truncated projections), 3D
tomography from Orlov and Tuy conditions to Kolsher filter and Katsevich
reconstruction formula, and dynamic tomography.
In the introduction we present briefly the physical interactions between photons and
matter. We derive the mathematical problem formulation of a function reconstruction
from its projections. We introduce the Radon transform and the xray transform, their
basic properties, in particular the Fourier slice theorem. In 2D tomography, we
demonstrate the Filtered Back Projection inversion formula and its application to fan
beam geometries. We then concentrate on recent advances in ROI reconstruction from
incomplete projections. In 3D reconstruction, we derive inversion conditions and
formulas for the parallel geometry and the Cone Beam Geometry. We develop recent
advances based on the Katsevich formula. We then introduce dynamics problems, i.e.
reconstruction from dynamic objects. We consider the problem reconstructing a 2D
dynamic object from its projections and show extensions to 3D dynamic reconstruction.

### (I) Visualisation scientifique (18h – G.P. Bonneau)

## (S) Kernel methods in machine learning

3ECTS 18h, Teacher: Zaid Harchaoui (zaid.harchaoui@gmail.com)

Objectives: To provide an introduction to kernel methods in Statistics for machine
learning.
Prerequisites: The minimal prerequisites for this course are a mastering of basic
Probability theory for discrete and continuous variables and of basic Statistics.
Schedule: to be precised.
Textbooks/references: to be precised.

## (S) Computational statistics in biology and medicine

3ECTS, (C 18H), Lecturers: O. François and JB Durand/G Bouchard

Corresponding to the french UE (Ensimag): « Algorithmes et statistique »
Computational statistics concerns the application of algorithmic techniques to problems
in statistics and in the analysis of large data sets. At the interface between computer
science and statistics, this field addresses a large number of applications in biology and
medicine. This course will give an overview of methodological and practical approaches
in computational statistics with applications to epidemiology, genetics or medical signal
analysis.

## (S) Point processes, reliability and survival analysis

3ECTS, O Gaudoin

Random point processes are used for modeling the occurrence of recurrent events in
time. Their study has multiple applications in the areas of health, industry, demography,
actuarial science, etc… The objective of this course is to present the essentials of
stochastic modeling and statistical inference for such processes. The preferred fields of
application will be reliability and survival analysis.

## (S) Time Series Analysis

3ECTS, A. Latour

This course attempts to provide a comprehensive introduction to time series analysis. It
gives an account of linear time series models and their application to the modeling and
forecasting of data collected sequentially in time. A time series is a sequence of random
variables Xt, the index t in Z being referred to as “time”. Typically the observations are
dependent and one aim is to predict the “future” given observations X1, . . . , Xn in the
“past”. Although the basic statistical concepts apply (such as likelihood, mean square
errors, etc.) the dependence must be taken into account. In the spirit of Brockwell and
Davis (1991), the approach is based on elementary Hilbert space methods.

### Méthodes de Monte-Carlo en finance (18h – J. Lelong et M. Echenim)

### Fondements mathématiques du calcul stochastique (36h – P. Etore)

### Gestion dynamique des risques financiers 1 (18h – P. Etoré)

### Calcul stochastique avancé (18h – P. Etoré)