# Convergent numerical approximation of the stochastic total variation flow

## TL;DR

This paper investigates the stochastic total variation flow with multiplicative noise, establishing existence of solutions via regularization, proposing an energy-preserving finite element method, and demonstrating convergence through numerical experiments.

## Contribution

It introduces a new finite element approximation for the stochastic total variation flow and proves its convergence to the true solution, addressing regularization and numerical stability.

## Key findings

- Numerical solutions converge to the true stochastic total variation flow.
- The proposed method preserves energy in the discretization.
- Numerical experiments confirm the practicability of the approach.

## Abstract

We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter $\varepsilon$ we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully discrete finite element approximation for the regularized gradient flow equation and show that the numerical solution converges to the solution of the unregularized STVF equation. We perform numerical experiments to demonstrate the practicability of the proposed numerical approximation. This paper contains a mistake: in the proof of Lemma 4.4 the last inequality is not valid. Meanwhile, this mistake has been fixed in [6] for a slightly modified numerical approximation in spatial dimension $d=1$. For $d\geq 1$ the validity of the estimate in Lemma 4.4 is still open but the convergence of the numerical approximation can be shown by a different approach, see [5].