La recherche

We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.

This thesis studies the persistent homology of real-valued continuous functions f on compact topological spaces X. The introduction of homological indices and homological dimensions allows us to link persistence theory to metric quantities of the compact space X, such as its upper-box dimension. These quantities give a precise framework to the Wasserstein p-stability results known in the literature, but also extend them to Hölder functions on more general spaces (including all compact Riemannian manifolds) with explicit constants and whose regime for p is optimal. In degree zero of homology, a more in-depth study can be made using trees associated to f, which generalize the merge trees which are definable when f is Morse. It is possible to link the dimension of these trees to the persistence index of f and to its barcode. We apply these deterministic results to the stochastic setting to draw consequences about the barcodes of random functions of prescribed regularity. These consequences also allow us to develop distributional discrimination tests for the processes, of which we present a particular example. Finally, we define the zeta-functions associated with a stochastic process and compute these functions and other related quantities for several processes in dimension one, including the Brownian motion and the alpha-stable Lévy processes.