We derive a posteriori error estimates for a fully discrete time-implicit finite element approximation of the stochastic total variaton flow (STVF) with additive space time noise. The estimates are first derived for an implementable fully discrete approximation of a regularized stochastic total variation flow. We then show that the derived a posteriori estimates remain valid for the unregularized flow up to a perturbation term that can be controlled by the regularization parameter. Based on the derived a posteriori estimates we propose a pathwise algorithm for the adaptive space-time refinement and perform numerical simulation for the regularized STVF to demonstrate the behavior of the proposed algorithm.

翻译：我们得出一个完全离散的、时间不限的参数近似值,该参数带有添加空间时间噪音;该估计数首先用于一个可执行的、完全离散的、定期的、随机的、总变异流;然后我们表明,从后推估计值对于可受正规化参数控制的到扰动期的不正规流动仍然有效;根据由此得出的一个事后估计值,我们提出一个适应性空间时间改进的路径算法,并对正规化的STVF进行数字模拟,以显示拟议算法的行为。