In recent years, SPDEs have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes are a powerful tool for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where we split the domain into sub-domains. Instead of solving one expensive problem on the entire domain, we deal with cheaper problems on the sub-domains. This is particularly useful in modern computer architectures, as the sub-problems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.

翻译：近年来，随机偏微分方程已成为数学中一个研究深入的领域。随着其日益流行，高效逼近其解变得尤为重要。因此，我们的目标是为随机偏微分方程发展高效实用的时间步进方法做出贡献。算子分裂格式是处理确定性和随机微分方程的有力工具。区域分解格式正是其中一例，我们将计算域划分为若干子域。相比在整个域上求解一个高成本问题，我们处理子域上的廉价问题。这在现代计算机架构中尤其有用，因为子问题通常可以并行求解。尽管分裂方法已被用于研究确定性偏微分方程的区域分解方法，但这在处理随机偏微分方程方面尚属新途径。我们为随机演化方程的分裂格式提供抽象收敛性分析，并给出区域分解格式作为该框架的应用实例。理论结果通过数值实验得到了验证。