标题： 随机全变分流的概率弱解和强解的数值逼近 显示英文标题

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

摘要： 我们提出了一种完全实用的数值方案，用于模拟随机总变差流（STFV）。 该近似基于正则化STVF方程的稳定时间隐式有限元时空逼近。 该近似还涉及噪声的有限维离散化，使该方案能够在物理硬件上完全实现。 我们证明了所提出的数值方案收敛到在随机变分不等式（SVIs）意义下定义的解。 作为我们收敛分析的副产品，我们提供了随机偏微分方程（SPDEs）的概率弱解概念到SVI设置的一般化。 我们还证明了如果路径唯一性成立，该数值方案在概率上收敛到一个概率强解。 我们进行了数值模拟，以说明所提出的数值方案{以及其非符合变体}在图像去噪背景下的行为。 显示英文摘要

摘要： We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STFV). 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 to a solution that is defined in the sense of stochastic variational inequalities (SVIs). 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.