**Objective**:

An inverse problem occurs when one tries to determine the

**internal state**of a system from measurements of the system, whose structure is assumed known. Inverse problems are quite frequent in science and engineering, as soon as information is required about a system, that cannot be measured directly.

Examples of inverse problems are provided by all

**medical imaging**techniques, and can also be found in

**earth sciences**(seismic prospection),

**astronomy**(noisy image restoration), or

**finance**(volatility calibration).

The aim of the course is to first introduce, through various examples, the

**origins**of inverse problems, to show their

**instability**, to present

**methods**for

**analyzing**these problems and to give some

**tools**that enable to solve inverse problem, and to quantify the quality of a solution. The course will show what is a

**regularization**method, how it can be used, and will highlight the fundamental

**balance**between

**stability and accuracy.**It will introduce numerical tools for analyzing inverse problems, such as the

**singular value decomposition**, and the

**adjoint state**method.

These concepts will be illustrated by practical training sessions.

**Programme**:

**Introduction**: origin of inverse problems, examples (integral equations)**Linear models**: least squares, singular value decomposition**Regularization**: Tikhonov method, a-priori and a-posteriori strategies**Statistics**: regression analysis, Bayesian estimation**Non-linear models**: parameter, state, observation, link with optimization**Adjoint state**: gradient computation, differential equations, parametrization**Project**

Each half-day session will feature 2 main lectures, and one exercise or practice session.

**Requirements :**Linear algebra, multi-variable calculus

**Last Modification :**Wednesday 11 July 2012