Numerical Approximation of Nonlinear SPDE's

TL;DR

This paper surveys numerical methods for solving nonlinear stochastic parabolic PDEs, integrating deterministic PDE approximation techniques with stochastic ODE methods to provide a unified framework accessible to researchers in both fields.

Contribution

It unifies and consolidates recent theoretical developments into a cohesive framework for approximating nonlinear SPDEs, bridging deterministic and stochastic numerical methods.

Findings

Provides a comprehensive survey of numerical techniques for nonlinear SPDEs.

Introduces a unified approach combining deterministic and stochastic approximation methods.

Includes examples demonstrating the applicability of the integrated framework.

Abstract

The numerical analysis of stochastic parabolic partial differential equations of the form $$ du + A(u) = f \,dt + g \, dW, $$ is surveyed, where $A$ is a partial operator and $W$ a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory.