Numerical approximation of the stochastic total variation flow

Martin Ondreját (Czech Academy of Sciences, Prague)

Feb 14. 2024, 11:00 — 11:40

The talk is based on a joint work with L. Baňas (Bielefeld). We propose fully practical numerical schemes (conformal and non-conformal) for the simulation of the stochastic total variation flow (STVF). The approximation is based on a stable time-implicit finite element space-time approximation of a regularized STVF equation. The approximation also involves a finite dimensional discretization of the noise that makes the scheme fully implementable on physical hardware. We show that the proposed numerical scheme converges in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. As a by product of our convergence analysis we provide a generalization of the concept of probabilistically weak solutions of stochastic partial differential equation (SPDEs) to the setting of SVIs. We also prove convergence of the numerical scheme to a probabilistically strong solution in probability if pathwise uniqueness holds. We perform numerical simulations to illustrate the behavior of the proposed numerical scheme as well as its non-conforming variant in the context of image denoising.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Stochastic Partial Differential Equations (Workshop)
Sandra Cerrai (U of Maryland)
Martin Hairer (Imperial College London)
Carlo Marinelli (University College London)
Eulalia Nualart (U Barcelona)
Luca Scarpa (Politecnico Milano)
Ulisse Stefanelli (U of Vienna)