论文标题
随机总变化流的概率弱和强溶液的数值近似
Numerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow
论文作者
论文摘要
我们提出了一个完全实用的数值方案，以模拟随机总变化流（STFV）。近似值基于正规化STVF方程的稳定时空有限元时空近似。近似值还涉及噪声的有限维度离散化，这使得该方案可以在物理硬件上完全实现。我们表明，所提出的数值方案会收敛到以随机变化不平等（SVI）意义定义的解决方案。作为我们的收敛分析的产物，我们提供了对SVIS设置的随机部分微分方程（SPDE）的概率弱解决方案的概念的概括。如果路径唯一性保持，我们还证明了数值方案与概率上强的解决方案的收敛性。我们执行数值模拟，以说明在图像denoising的上下文中提出的数值方案及其不合格变体的行为。
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.
