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

TL;DR

This paper introduces a practical finite element scheme for simulating the stochastic total variation flow, proving convergence to solutions in the framework of stochastic variational inequalities and demonstrating its application in image denoising.

Contribution

It develops a fully implementable numerical scheme for STVF, extending the concept of weak solutions to SVIs and proving convergence to strong solutions under certain conditions.

Findings

Scheme converges to SVI solutions

Numerical simulations demonstrate effectiveness in image denoising

Generalizes probabilistic weak solutions for SPDEs

Abstract

We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STFV). 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 to a solution that is defined in the sense of stochastic variational inequalities (SVIs). 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.