Das Cauchy-Problem bei quasilinearen parabolischen Differentialgleichungen mit unstetigen Anfangswerten.
Deep learning for gradient flows using the Brezis–Ekeland principle
We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.
Dépendance continue de solutions généralisées locales
Description of regional blow-up in a porous-medium equation.
Diffusion of a particle in presence of N moving point sources
Discontinuous parabolic problems with a nonlocal initial condition.
Discrete maximum principle in parabolic boundary-value problems
Discretization methods with analytical characteristic methods for advection-diffusion-reaction equations and 2d applications
Our studies are motivated by a desire to model long-time simulations of possible scenarios for a waste disposal. Numerical methods are developed for solving the arising systems of convection-diffusion-dispersion-reaction equations, and the received results of several discretization methods are presented. We concentrate on linear reaction systems, which can be solved analytically. In the numerical methods, we use large time-steps to achieve long simulation times of about 10 000 years. We propose...
Distribution-valued weak solutions to a parabolic problem arising in financial mathematics.
Dynamique des tourbillons de vorticité pour l’équation de Ginzburg-Landau parabolique