Numerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow

¹ Department of Mathematics, Bielefeld University, 33501 Bielefeld, Germany
² Institute of Information Theory and Automation, Pod Vodárenskou vĕž 4, CZ-182 00 Praha 8, Czech Republic

^* Corresponding author: banas@math.uni-bielefeld.de

Received: 4 May 2022
Accepted: 1 November 2022

Abstract

We propose a fully practical numerical scheme 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.