Convergent numerical approximation of the stochastic total variation flow

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].