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Teaching coordinator :

Michel Kern

Level : Graduate

Course Language : French

Term : Fall

Number of hours : 26

ECTS Credits : 2
SGS_S1923 Inverse problems
Teaching site :
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.

  • 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

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