SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024.
Abstract. In recent years, stochastic partial differential equations (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 provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems 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. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. 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.

中文翻译：

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随机进化方程的域分解方法


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2611-2639 页，2024 年 12 月。

抽象。近年来，随机偏微分方程 （SPDE） 已成为数学中一个研究范围很广的领域。随着它们的日益普及，有效地近似他们的解决方案变得很重要。因此，我们的目标是为 SPDE 开发高效实用的时间步长方法做出贡献。运算符拆分方案为确定性和随机微分方程提供了强大、高效且灵活的数值方法。域分解方案给出了一个例子，其中将域拆分为子域，并在分而治之策略中构建数值近似。不是解决整个域上的一个昂贵的问题，而是处理子域上的更便宜的问题。这在现代计算机体系结构中特别有用，因为子问题通常可以并行解决。虽然已经使用分裂方法研究确定性 PDE 的域分解方法，但这是 SPDE 的一种新方法。这意味着现有的收敛分析并不直接适用，即使算子分裂域分解方法的构建块是标准的。我们提供了随机进化方程的分裂方案的抽象收敛分析，并陈述了域分解方案作为该设置的应用。通过数值实验验证了理论结果。