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A DOMAIN DECOMPOSITION METHOD FOR STOCHASTIC EVOLUTION EQUATIONS

Buckwar, Evelyn LU ; Djurdjevac, Ana and Eisenmann, Monika LU orcid (2024) In SIAM Journal on Numerical Analysis 62(6). p.2611-2639
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... (More)

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.

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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
domain decomposition, operator splitting, stochastic partial differential equations
in
SIAM Journal on Numerical Analysis
volume
62
issue
6
pages
29 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:85210912150
ISSN
0036-1429
DOI
10.1137/24M1629845
language
English
LU publication?
yes
id
38c04cf7-e771-455d-9f01-15c2a02d625b
date added to LUP
2025-01-27 14:33:33
date last changed
2025-04-04 15:20:13
@article{38c04cf7-e771-455d-9f01-15c2a02d625b,
  abstract     = {{<p>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.</p>}},
  author       = {{Buckwar, Evelyn and Djurdjevac, Ana and Eisenmann, Monika}},
  issn         = {{0036-1429}},
  keywords     = {{domain decomposition; operator splitting; stochastic partial differential equations}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{2611--2639}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Numerical Analysis}},
  title        = {{A DOMAIN DECOMPOSITION METHOD FOR STOCHASTIC EVOLUTION EQUATIONS}},
  url          = {{http://dx.doi.org/10.1137/24M1629845}},
  doi          = {{10.1137/24M1629845}},
  volume       = {{62}},
  year         = {{2024}},
}