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  4. Impact of spatial coarsening on Parareal convergence
 
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Impact of spatial coarsening on Parareal convergence

Citation Link: https://doi.org/10.15480/882.4311
Publikationstyp
Preprint
Date Issued
2021-11-19
Sprache
English
Author(s)
Angel, Judith  orcid-logo
Götschel, Sebastian  orcid-logo
Ruprecht, Daniel  orcid-logo
Institut
Mathematik E-10  
TORE-DOI
10.15480/882.4311
TORE-URI
http://hdl.handle.net/11420/12372
Citation
arXiv: 2111.10228 (2021)
Publisher DOI
10.48550/arXiv.2111.10228v1
ArXiv ID
2111.10228v1
We study the impact of spatial coarsening on the convergence of the Parareal algorithm, both theoretically and numerically. For initial value problems with a normal system matrix, we prove a lower bound for the Euclidean norm of the iteration matrix. When there is no physical or numerical diffusion, an immediate consequence is that the norm of the iteration matrix cannot be smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom than the fine. This prevents a theoretical guarantee for monotonic convergence, which is necessary to obtain meaningful speedups. For diffusive problems, in the worst-case where the iteration error contracts only as fast as the powers of the iteration matrix norm, making Parareal as accurate as the fine method will take about as many iterations as there are processors, making meaningful speedup impossible. Numerical examples with a non-normal system matrix show that for diffusive problems good speedup is possible, but that for non-diffusive problems the negative impact of spatial coarsening on convergence is big.
Subjects
Mathematics - Numerical Analysis
Computer Science - Computational Engineering; Finance; and Science
Computer Science - Numerical Analysis
DDC Class
510: Mathematik
Funding(s)
TIME parallelisation: for eXascale computing and beyond  
Funding Organisations
Bundesministerium für Bildung und Forschung (BMBF)  
European High-Performance Computing Joint Undertaking (JU)
Lizenz
http://rightsstatements.org/vocab/InC/1.0/
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