A BOOTSTRAP MULTIGRID EIGENSOLVER

James Brannick; Shuhao Cao

doi:10.1137/20M131151X

A BOOTSTRAP MULTIGRID EIGENSOLVER

Research output: Contribution to journal › Article › peer-review

Abstract

This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid setup algorithm that includes an AMG eigensolver, we illustrate how the algorithm can be simplified for the case of a discretized partial differential equation, thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace–Beltrami eigenvalue problem.

Original language	English (US)
Pages (from-to)	1627-1657
Number of pages	31
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	43
Issue number	4
DOIs	https://doi.org/10.1137/20M131151X
State	Published - 2022

All Science Journal Classification (ASJC) codes

Analysis

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10.1137/20M131151X

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title = "A BOOTSTRAP MULTIGRID EIGENSOLVER",

abstract = "This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid setup algorithm that includes an AMG eigensolver, we illustrate how the algorithm can be simplified for the case of a discretized partial differential equation, thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace–Beltrami eigenvalue problem.",

author = "James Brannick and Shuhao Cao",

note = "Funding Information: ∗Received by the editors January 8, 2020; accepted for publication (in revised form) by J. H. Adler June 28, 2022; published electronically November 18, 2022. https://doi.org/10.1137/20M131151X Funding: The authors{\textquoteright} research was supported in part by the National Science Foundation under grants DMS-1320608, DMS-1418934, DMS-1620346, and DMS-2136075. †Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802 USA (brannick@psu.edu). ‡Division of Computing, Analytics, and Mathematics, School of Science and Engineering, University of Missouri Kansas City, Kansas City, MO 64110 USA (sch59@umkc.edu). Publisher Copyright: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

year = "2022",

doi = "10.1137/20M131151X",

language = "English (US)",

volume = "43",

pages = "1627--1657",

journal = "SIAM Journal on Matrix Analysis and Applications",

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N1 - Funding Information: ∗Received by the editors January 8, 2020; accepted for publication (in revised form) by J. H. Adler June 28, 2022; published electronically November 18, 2022. https://doi.org/10.1137/20M131151X Funding: The authors’ research was supported in part by the National Science Foundation under grants DMS-1320608, DMS-1418934, DMS-1620346, and DMS-2136075. †Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802 USA (brannick@psu.edu). ‡Division of Computing, Analytics, and Mathematics, School of Science and Engineering, University of Missouri Kansas City, Kansas City, MO 64110 USA (sch59@umkc.edu). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.

PY - 2022

Y1 - 2022

N2 - This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid setup algorithm that includes an AMG eigensolver, we illustrate how the algorithm can be simplified for the case of a discretized partial differential equation, thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace–Beltrami eigenvalue problem.

AB - This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid setup algorithm that includes an AMG eigensolver, we illustrate how the algorithm can be simplified for the case of a discretized partial differential equation, thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace–Beltrami eigenvalue problem.

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A BOOTSTRAP MULTIGRID EIGENSOLVER

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