A posteriori estimates for the stochastic total variation flow

TL;DR

This paper develops a posteriori error estimates for a fully discrete approximation of the stochastic total variation flow, enabling adaptive refinement and improved numerical simulations of the flow under stochastic influences.

Contribution

It introduces a posteriori error estimates for the stochastic total variation flow and proposes an adaptive algorithm based on these estimates for better numerical approximation.

Findings

Error estimates are valid for regularized and unregularized flows.

The adaptive algorithm improves the accuracy of numerical simulations.

Numerical experiments demonstrate the effectiveness of the proposed method.

Abstract

We derive a posteriori error estimates for a fully discrete time-implicit finite element approximation of the stochastic total variaton flow (STVF) with additive space time noise. The estimates are first derived for an implementable fully discrete approximation of a regularized stochastic total variation flow. We then show that the derived a posteriori estimates remain valid for the unregularized flow up to a perturbation term that can be controlled by the regularization parameter. Based on the derived a posteriori estimates we propose a pathwise algorithm for the adaptive space-time refinement and perform numerical simulation for the regularized STVF to demonstrate the behavior of the proposed algorithm.