INVERSE PROBLEMS

The objectives of this course is to learn and undestand various way to solve inverse problems. Depending on the student's area, applications wil be oriented toward photographic 3D-reconstruction methods or numerial problems with uncertainty. In the first, case, the problem is to obtain a 3D model of a scene i.e., its shape and its colour.In the second case, the main filtering methods based on the non-linear Bayesian filters (particle filter, Kalman filter, extended Kalman filter, ensemble Kalman filter) will be studied. For a given ODE/SDE, students have to identify the corresponding notion of integration, then should be able to propose an adapted filtering method.

The content is twofold, with a focus on the student's preferred area:

- Filtering methods

    - Introduction to filtering : Bayesian inference ; Filtering and smoothing principles, non-linear filtering ; Application to the linear and Gaussian case: Kalman filter.

    - Uncertainty dynamics for ordinary differential equations (ODE) and stochastic differential equations (SDE): from partial differential equation to ODE (numerical schemes); Lyapunov exponent and chaotic system; stochastic processes; discrete/continuous Markov processes; Observable/measure dynamics duality

     - Stochastic filtering: Particle filter; Ensemble Kalman filter; Stochastic smoother

Optimization, notion of probability and statistics, numerical linear algebra