Numerical approximation of viscosity solutions of Hamilton-Jacobi equations and example.

Summary This thesis concerns the numerical approximation of the viscosity solutions, as defined by Michael Grain Crandall and Pierre-Louis Lions, of the first order Hamilton-Jacobi equations which are nonlinear partial differential equations, as well as the study of an example from image processing, shape-from-shading, which consists in the reconstruction of a three-dimensional relief from the data of a two-dimensional image, a photograph for example. The first chapter is a brief presentation of the viscosity solutions of the Hamilton-Jacobi equations and of some existence and uniqueness results. The second chapter describes the different methods developed to approximate these solutions, and is based on numerical analysis. The third chapter, more applied, aims at explaining how, concretely, one can write an approximation scheme for viscosity solutions. Finally, the example is studied in a precise way (by taking up the different developments of the first chapters of the thesis): we show how the relief can be interpreted as the viscosity solution of a Hamilton-Jacobi equation. We study the different possible formalizations for the edges of the image in order to reach satisfactory existence and uniqueness results. Then a scheme is constructed and applied to the numerical reconstruction of different images.