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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985556%3A_____%2F23%3A00571182" target="_blank" >RIV/67985556:_____/23:00571182 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.esaim-m2an.org/articles/m2an/abs/2023/02/m2an220087/m2an220087.html" target="_blank" >https://www.esaim-m2an.org/articles/m2an/abs/2023/02/m2an220087/m2an220087.html</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1051/m2an/2022089" target="_blank" >10.1051/m2an/2022089</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10103 - Statistics and probability

Projekt

<a href="/cs/project/GA22-12790S" target="_blank" >GA22-12790S: Stochastické systémy v nekonečné dimensi</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

ESAIM. Mathematical Modelling and Numerical Analysis

ISSN

2822-7840

e-ISSN

2804-7214

Svazek periodika

57

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

FR - Francouzská republika

Počet stran výsledku

31

Strana od-do

785-815

Kód UT WoS článku

000959169100009

EID výsledku v databázi Scopus

2-s2.0-85142299302