|arXiv ID:||2111.10228v1||Title:||Impact of spatial coarsening on Parareal convergence||Language:||English||Authors:||Angel, Judith
|Keywords:||Mathematics - Numerical Analysis; Mathematics - Numerical Analysis; Computer Science - Computational Engineering; Finance; and Science; Computer Science - Numerical Analysis||Issue Date:||19-Nov-2021||Source:||arXiv: 2111.10228v1 (2021)||Abstract (english):||
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.
|URI:||http://hdl.handle.net/11420/12372||DOI:||10.15480/882.4311||Institute:||Mathematik E-10||Document Type:||Article||Project:||TIME parallelisation: for eXascale computing and beyond
|Funded by:||Bundesministerium für Bildung und Forschung (BMBF)
European High-Performance Computing Joint Undertaking (JU)
|More Funding information:||This project has received funding from the European High-Performance Computing Joint Undertaking (JU) under grant agreement No 955701. The JU receives support from the European Union's Horizon 2020 research and innovation programme and Belgium, France, Germany, and Switzerland. This project also received funding from the German Federal Ministry of Education and Research (BMBF) grant 16HPC048. The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 \Modeling, Simulation and Optimization of Fluid Dynamic Applications.||License:||In Copyright|
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