A hybrid probabilistic domain decomposition algorithm suited for very large-scale elliptic PDEs

State of the art domain decomposition algorithms for large-scale boundary value problems (with \(M\gg 1\) degrees of freedom) suffer from bounded strong scalability because they involve the synchronisation and communication of workers inherent to iterative linear algebra. Here, we introduce PDDSpars...

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Bibliographic Details
Published inarXiv.org
Main Authors Bernal, Francisco, Morón-Vidal, Jorge, Acebrón, Juan A
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 13.01.2023
Subjects
Algorithms
Boundary value problems
Computer Science - Numerical Analysis
Domain decomposition methods
Fault tolerance
Iterative methods
Linear algebra
Mathematical analysis
Mathematics - Numerical Analysis
Synchronism
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Summary:State of the art domain decomposition algorithms for large-scale boundary value problems (with \(M\gg 1\) degrees of freedom) suffer from bounded strong scalability because they involve the synchronisation and communication of workers inherent to iterative linear algebra. Here, we introduce PDDSparse, a different approach to scientific supercomputing which relies on a "Feynman-Kac formula for domain decomposition". Concretely, the interfacial values (only) are determined by a stochastic, highly sparse linear system \(G(\omega){\vec u}={\vec b}(\omega)\) of size \({\cal O}(\sqrt{M})\), whose coefficients are constructed with Monte Carlo simulations-hence embarrassingly in parallel. In addition to a wider scope for strong scalability in the deep supercomputing regime, PDDSparse has built-in fault tolerance and is ideally suited for GPUs. A proof of concept example with up to 1536 cores is discussed in detail.
ISSN:2331-8422
DOI:10.48550/arxiv.2301.05780