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

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 allow 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 $2.1\times$ and $3.3\times$ for Laplace problems and between $1.7\times$ and $4.0\times$ 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, \href{https://github.com/barbagroup/fmm-bem-relaxed}{https://github.com/barbagroup/fmm-bem-relaxed}.

21 pages, 20 figures. Second version submitted for peer review on March 2016, with all results re-computed and revised author list. Rejected in October 2016. Currently undergoing revision for a third submission. See progress of open revision in https://github.com/barbagroup/inexact-gmres

References in corpus (2)

• Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUs and a billion unknowns
• A Tuned and Scalable Fast Multipole Method as a Preeminent Algorithm for Exascale Systems

Cited by in corpus (1)

• fmmgen: Automatic Code Generation of Operators for Cartesian Fast Multipole and Barnes-Hut Methods