Abstract

Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a “minimax” problem, minimizing with respect to solver parameters the appropriate measure of solver work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

Keywords

  1. local Fourier analysis
  2. minimax problem
  3. multigrid methods
  4. robust optimization

MSC codes

  1. 47N40
  2. 65M55
  3. 90C26
  4. 49Q10

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Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis

Authors: Jed Brown, Yunhui He, Scott MacLachlan, Matt Menickelly, Stefan M. Wild

File: supp.pdf

Type: PDF

Contents: LFA Symbols are presented for components of the multigrid algorithms that are not given in the text or easily found in the cited literature. Additionally, details of the software interface used for the experiments are presented.

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A109 - A138
ISSN (online): 1095-7197

History

Submitted: 23 December 2019
Accepted: 17 September 2020
Published online: 5 January 2021

Keywords

  1. local Fourier analysis
  2. minimax problem
  3. multigrid methods
  4. robust optimization

MSC codes

  1. 47N40
  2. 65M55
  3. 90C26
  4. 49Q10

Authors

Affiliations

Funding Information

Office of Science https://doi.org/10.13039/100006132 : DE-SC0016140, DE-AC02-06CH11357
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038

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