Université Paris 13 MACS 3 - Advanced Numerical Analysis Professors : Marion Darbas and Caroline Japhet

This course presents the analysis of mathematical, theoretical, and numerical methods for solving complex problems (partial differential equations, ill-posed problems, etc.). It draws on various applications, particularly in medicine, meteorology, and geosciences. We will focus in particular on domain decomposition methods, as well as methods for solving inverse problems. These approaches can be combined to take advantage of their complementarities. The program ins as follows :

I - Introduction to Domain Decomposition Methods

• Introduction and Examples of Applications
• Schwarz-type Methods, with and without Overlap (Parallel or Alternating) 
• Study at the Continuous Level (Convergence, Importance of Transmission Conditions) 
• Writing at the Discrete Level ▷ Interface Problem, Preconditioning ▷ Applications (Meteorology/Porous Media): Model Couplings
  

II - Introduction to Inverse Problems

• Part I - General Information : Direct Problems and Inverse Problems / Examples of Inverse Problems / Well- and Ill-Posed Problems 
• Part II - Solution Methods : Linear Inverse Problems - Least Squares - Conditioning / Minimization Methods - Computing the Adjoint State / Regularization Methods 
• Part III - Medical Application: Inverse Problem of Sources in EEG
  

Handouts

• Numerical Analysis (François Cuvelier)
  

Cours

Diapositives du cours, devoir et TD/TP corrigés : voir le site où a lieu le cours

References

• F. Cuvelier. Cours d’Analyse Numérique Elémentaire, MACS 1, 2020.
• M.J. Gander and L. Halpern. Méthodes de décomposition de domaines - Notions de base. In Encyclopédie
  des techniques de l’ingénieur, méthodes numériques. AF137, 2012.
• P.-L. Lions. On the Schwarz alternating method. I. In R. Glowinski, G. H. Golub, G. A. Meurant, and J. Pé-
  riaux, editors, First International Symposium on Domain Decomposition Methods for Partial Differential
  Equations, pages 1–42. Philadelphia, PA, SIAM, 1988.
• H. A. Schwarz. Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Natur-
  forschenden Gesellschaft in Zürich, 15 :272–286, May 1870.