- 5MMMNA

Goal(s)

The goal of this course is to present and analyze a wide range of numerical methods and algorithms that find applications in various modeling fields. The first part is dedicated to the numerical resolution of partial differential equations (PDEs) and the second one to numerical methods for optimal transport problems.

Content(s)

In the first part, we will start by reminding the principle of the finite difference method (FDM) and the finite element method (FEM). Then, they will be compared to the finite volume method (FVM), a method well suited for the numerical simulation of various conservation laws. Also, more advanced type of finite element methods will be considered (e.g. the mixed finite element method or the discontinuous Galerkin method) in order to solve efficiently a larger range of problems. To test these methods, the numerical resolution of convection-diffusion problems will be discussed, with a potential focus on drift-diffusion models used to describe the electron flow in semiconductor devices.

The second part deals with the optimal transport theory. It is an important field of mathematics that was originally introduced in the 1700’s by the French mathematician and engineer Gaspard Monge. This theory has connections with PDEs, geometry and probability and has been used in many fields such as computer vision, economy, non-imaging optics… In the last 15 years, this problem has been extensively studied from a computational point of view and different efficient algorithms have been proposed.
In this part, we present the analysis of several algorithms using the notion of duality, such as Auction’s algorithm, Sinkorn algorithm, Oliker-Prüssner algorithm and a Newton algorithm.

The first part is based on the 2A courses «Partial differential equations and numerical methods » and « Variational methods applied to modelling ». Someone who did not follow these courses needs to bring up to date PDE knowledges.