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Title: A new adaptive GMRES algorithm for achieving high accuracy

Abstract

GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram-Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES (k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. The essence of the adaptive GMRES strategy is to adapt the parameter k to the problem, similar in spirit to how a variable order ODE algorithm tunes the order k. With FORTRAN 90, which provides pointers and dynamic memory management, dealing with the variable storage requirements implied by varying k is not too difficult. The parameter k can be both increased and decreased-an increase-only strategy is described next followed by pseudocode.

Authors:
; ;  [1];  [2]
  1. Virginia Polytechnic Inst., Blacksburg, VA (United States)
  2. Utah State Univ., Logan, UT (United States)
Publication Date:
Research Org.:
Front Range Scientific Computations, Inc., Lakewood, CO (United States)
OSTI Identifier:
433394
Report Number(s):
CONF-9604167-Vol.1
Journal ID: ISSN 1070--5325; ON: DE96015306; CNN: Grant F49620-92-J-0236; TRN: 97:000720-0069
DOE Contract Number:  
FG05-88ER25068
Resource Type:
Conference
Resource Relation:
Journal Volume: 5; Journal Issue: 4; Conference: Copper Mountain conference on iterative methods, Copper Mountain, CO (United States), 9-13 Apr 1996; Other Information: PBD: [1996]; Related Information: Is Part Of Copper Mountain conference on iterative methods: Proceedings: Volume 1; PB: 422 p.
Country of Publication:
United States
Language:
English
Subject:
99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; ALGORITHMS; PERFORMANCE; ACCURACY; CONVERGENCE; FORTRAN; ITERATIVE METHODS; MEMORY MANAGEMENT; LINEAR PROGRAMMING

Citation Formats

Sosonkina, M, Watson, L T, Kapania, R K, and Walker, H F. A new adaptive GMRES algorithm for achieving high accuracy. United States: N. p., 1996. Web. doi:10.1002/(SICI)1099-1506(199807/08)5:4<275::AID-NLA131>3.0.CO;2-B.
Sosonkina, M, Watson, L T, Kapania, R K, & Walker, H F. A new adaptive GMRES algorithm for achieving high accuracy. United States. https://doi.org/10.1002/(SICI)1099-1506(199807/08)5:4<275::AID-NLA131>3.0.CO;2-B
Sosonkina, M, Watson, L T, Kapania, R K, and Walker, H F. 1996. "A new adaptive GMRES algorithm for achieving high accuracy". United States. https://doi.org/10.1002/(SICI)1099-1506(199807/08)5:4<275::AID-NLA131>3.0.CO;2-B. https://www.osti.gov/servlets/purl/433394.
@article{osti_433394,
title = {A new adaptive GMRES algorithm for achieving high accuracy},
author = {Sosonkina, M and Watson, L T and Kapania, R K and Walker, H F},
abstractNote = {GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram-Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES (k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. The essence of the adaptive GMRES strategy is to adapt the parameter k to the problem, similar in spirit to how a variable order ODE algorithm tunes the order k. With FORTRAN 90, which provides pointers and dynamic memory management, dealing with the variable storage requirements implied by varying k is not too difficult. The parameter k can be both increased and decreased-an increase-only strategy is described next followed by pseudocode.},
doi = {10.1002/(SICI)1099-1506(199807/08)5:4<275::AID-NLA131>3.0.CO;2-B},
url = {https://www.osti.gov/biblio/433394}, journal = {},
issn = {1070--5325},
number = 4,
volume = 5,
place = {United States},
year = {1996},
month = {12}
}

Conference:
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