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A domain decomposition method for stochastic evolution equations

Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann

TL;DR

This work develops a domain-decomposition operator splitting framework for stochastic evolution equations, enabling parallel sub-domain solves through an IMEX Euler–Maruyama time-stepping scheme and a Lie-splitting assembly. It provides a rigorous variational-analytical foundation, including existence and regularity results, a well-posedness proof for the implicit subproblems, and a mean-square error bound that scales with the time-step size via $h^{2\nu}$ plus a truncation error $\varepsilon$. The domain-decomposition instantiation for a stochastic heat equation demonstrates how local sub-operators can be coupled to recover the global solution with controlled accuracy. Numerical experiments in 2D validate the theoretical convergence rate (approximately $\tfrac{1}{2}$) for both additive and multiplicative noise, highlighting the method’s practicality and parallelizability for SPDEs.

Abstract

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.

A domain decomposition method for stochastic evolution equations

TL;DR

This work develops a domain-decomposition operator splitting framework for stochastic evolution equations, enabling parallel sub-domain solves through an IMEX Euler–Maruyama time-stepping scheme and a Lie-splitting assembly. It provides a rigorous variational-analytical foundation, including existence and regularity results, a well-posedness proof for the implicit subproblems, and a mean-square error bound that scales with the time-step size via plus a truncation error . The domain-decomposition instantiation for a stochastic heat equation demonstrates how local sub-operators can be coupled to recover the global solution with controlled accuracy. Numerical experiments in 2D validate the theoretical convergence rate (approximately ) for both additive and multiplicative noise, highlighting the method’s practicality and parallelizability for SPDEs.

Abstract

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.
Paper Structure (17 sections, 9 theorems, 85 equations, 2 figures)

This paper contains 17 sections, 9 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analytical background
  3. Assumptions
  4. Solution concept and additional regularity
  5. A semi-discrete operator splitting scheme
  6. Derivation of method
  7. Well-posedness of the implicit equation
  8. Application: Domain decomposition
  9. Error analysis
  10. Boundedness of the numerical solution
  11. Regularity of the domain decomposition
  12. Error bound
  13. Numerical experiment
  14. Experiment 1
  15. Experiment 2
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let Assumptions ass:initialValue--ass:B be fulfilled. Then the stochastic evolution equation eq:StochEvEq has a unique (up to $\mathcal{P}$-indistinguishability) variational solution that fulfills

Figures (2)

  • Figure 1: Visuablization of the weight functions $\chi_1, \chi_2, \chi_3$ and $\chi_4$
  • Figure 2: Comparison of an IMEX-Euler--Maruyama scheme with and without a domain decomposition. The left-hand side shows the result of Experiment 1 and right-hand side the results of Experiment 2.

Theorems & Definitions (14)

  • Lemma 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Theorem 4.4: Bound for the error
  • ...and 4 more