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

0571182 - ÚTIA 2024 RIV FR eng J - Článek v odborném periodiku
Ondreját, Martin - Baňas, L.
Numerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow.
ESAIM. Mathematical Modelling and Numerical Analysis. Roč. 57, č. 2 (2023), s. 785-815. ISSN 2822-7840. E-ISSN 2804-7214
Grant CEP: GA ČR(CZ) GA22-12790S
Institucionální podpora: RVO:67985556
Klíčová slova: stochastic total variation flow * stochastic variational inequalities * image processing * finite element approximation * tightness in BV spaces
Obor OECD: Statistics and probability
Impakt faktor: 1.9, rok: 2022
Způsob publikování: Open access
http://library.utia.cas.cz/separaty/2023/SI/ondrejat-0571182.pdf https://www.esaim-m2an.org/articles/m2an/abs/2023/02/m2an220087/m2an220087.html

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.
Trvalý link: https://hdl.handle.net/11104/0342475