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Tuning Multigrid Methods with Robust Optimization and Local Fourier Analysis

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/19m1308669· OSTI ID:1765460
 [1];  [2];  [3];  [4];  [4]
  1. Univ. of Colorado, Boulder, CO (United States)
  2. Univ. of Waterloo, ON (Canada); Memorial Univ. of Newfoundland, St. John's, NL (Canada)
  3. Memorial Univ. of Newfoundland, St. John's, NL (Canada)
  4. Argonne National Lab. (ANL), Argonne, IL (United States)
+ Show Author Affiliations

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 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. Finally, several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

Research Organization:
Argonne National Laboratory (ANL)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Natural Sciences and Engineering Research Council of Canada (NSERC)
Grant/Contract Number:
AC02-06CH11357; SC0016140
OSTI ID:
1765460
Journal Information:
SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 1 Vol. 43; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
Local Fourier analysis
Minimax problem
Multigrid methods
Robust optimization