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Inexact GMRES iterations and relaxation strategies with fast-multipole boundary element method

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Abstract

Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, p. We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing p. In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between 1.5 × and 2.3 × for Laplace problems and between 2.7 × and 3.3 × for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, https://github.com/barbagroup/fmm-bem-relaxed, and we share reproducibility packages for all results in https://github.com/barbagroup/inexact-gmres/.

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Acknowledgements

This work was supported by the National Science Foundation via NSF CAREER award OCI- 1149784. LAB acknowledges support from NVIDIA Corp. via the CUDA Fellows Program. Dr. Cris Cecka (previously at Harvard University, currently at Nvidia Corp.) contributed to the development of the code, particularly writing the octree and base evaluator. He later continued developing his fast-multipole framework, which evolved into his FMMTL project (see https://github.com/ccecka/fmmtl). The authors also wish to acknowledge valuable interactions with Dr. Christopher Cooper (previously at Boston University, currently at Universidad Técnica Federico Santa María) that helped with the implementation of the boundary element method.

Funding

This work was supported by the National Science Foundation via NSF CAREER award OCI-1149784.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC, USA

    Tingyu Wang & Lorena A. Barba

  2. Nvidia, Corp., Santa Clara, CA, USA

    Simon K. Layton

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  1. Tingyu Wang
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Correspondence to Lorena A. Barba.

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Communicated by: Michael O’Neil

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This article belongs to the Topical Collection: Advances in Computational Integral Equations Guest Editors: Stephanie Chaillat, Adrianna Gillman, Per-Gunnar Martinsson, Michael O’Neil, Mary-Catherine Kropinski, Timo Betcke, Alex Barnett

Appendix: Algorithm listings

Appendix: Algorithm listings

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Wang, T., Layton, S.K. & Barba, L.A. Inexact GMRES iterations and relaxation strategies with fast-multipole boundary element method. Adv Comput Math 48, 32 (2022). https://doi.org/10.1007/s10444-022-09932-8

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