We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter \(\varepsilon \) we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully discrete finite element approximation for the regularized gradient flow equation and show that the numerical solution converges to the solution of the unregularized STVF equation. We perform numerical experiments to demonstrate the practicability of the proposed numerical approximation.

我们研究了带有线性乘性噪声的随机总变化流（STVF）方程。通过考虑相对于正则化参数\（\ varepsilon \）的正则化随机梯度流序列的极限，我们获得了满足随机变分不等式的STVF方程的唯一变分解。我们为正则化梯度流方程提出了一个能量守恒的完全离散有限元逼近，并证明了数值解收敛于非正则化STVF方程的解。我们进行数值实验，以证明所提出的数值近似的实用性。