Correction to: Convergent numerical approximation of the stochastic total variation flow

TL;DR

This paper corrects errors in a previous work on numerical approximation of the stochastic total variation flow, clarifying the solution definition and providing an alternative proof for a key estimate in one dimension, ensuring convergence of the scheme.

Contribution

It rectifies the definition of SVI solutions and offers an alternative proof for a discrete estimate in one dimension, improving the convergence analysis of the numerical scheme.

Findings

Corrected the boundary term in the SVI solution definition.

Provided an alternative proof of the discrete estimate in 1D.

Confirmed convergence of the numerical scheme in various dimensions.

Abstract

We correct two errors in our paper [4]. First error concerns the definition of the SVI solution, where a boundary term which arises due to the Dirichlet boundary condition, was not included. The second error concerns the discrete estimate [4, Lemma 4.4], which involves the discrete Laplace operator. We provide an alternative proof of the estimate in spatial dimension $d=1$ by using a mass lumped version of the discrete Laplacian. Hence, after a minor modification of the fully discrete numerical scheme the convergence in $d=1$ follows along the lines of the original proof. The convergence proof of the time semi-discrete scheme, which relies on the continuous counterpart of the estimate [4, Lemma 4.4], remains valid in higher spatial dimension. The convergence of the fully discrete finite element scheme from [4] in any spatial dimension is shown in [3] by using a different approach.