Fast multipole preconditioners for sparse matrices arising from elliptic equations

Huda Ibeid, Rio Yokota, Jennifer Pestana, David Keyes

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low-rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.
LanguageEnglish
Pages213-229
Number of pages17
JournalComputing and Visualization in Science
Volume18
Issue number6
Early online date9 Nov 2017
DOIs
Publication statusPublished - 31 Mar 2018

Fingerprint

Fast multipole Method
Sparse matrix
Preconditioner
Elliptic Equations
Supercomputers
Boundary Integral
Boundary element method
Boundary value problems
Scalability
Synchronization
Krylov Methods
Data storage equipment
Resolvent Operator
Global Synchronization
Matrix-vector multiplication
Precondition
Communication
Truncation Error
Multigrid Method
Supercomputer

Keywords

  • Fast multipole method
  • Preconditioner
  • Krylov subspace method
  • Poisson equation
  • Stokes equation

Cite this

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abstract = "Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low-rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.",
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Fast multipole preconditioners for sparse matrices arising from elliptic equations. / Ibeid, Huda; Yokota, Rio; Pestana, Jennifer; Keyes, David.

In: Computing and Visualization in Science, Vol. 18, No. 6, 31.03.2018, p. 213-229.

Research output: Contribution to journalArticle

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