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Journal Article

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

Ondreját Martin, Baňas L.

: ESAIM. Mathematical Modelling and Numerical Analysis vol.57, 2 (2023), p. 785-815

: GA22-12790S, GA ČR

: stochastic total variation flow, stochastic variational inequalities, image processing, finite element approximation, tightness in BV spaces

: 10.1051/m2an/2022089

: http://library.utia.cas.cz/separaty/2023/SI/ondrejat-0571182.pdf

: https://www.esaim-m2an.org/articles/m2an/abs/2023/02/m2an220087/m2an220087.html

(eng): We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STVF). 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 in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. 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.

: BA

: 10103