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 GramSchmidt 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 decreasedan increaseonly strategy is described next followed by pseudocode.
 Authors:

 Virginia Polytechnic Inst., Blacksburg, VA (United States)
 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):
 CONF9604167Vol.1
Journal ID: ISSN 10705325; ON: DE96015306; CNN: Grant F4962092J0236; TRN: 97:0007200069
 DOE Contract Number:
 FG0588ER25068
 Resource Type:
 Conference
 Resource Relation:
 Journal Volume: 5; Journal Issue: 4; Conference: Copper Mountain conference on iterative methods, Copper Mountain, CO (United States), 913 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)10991506(199807/08)5:4<275::AIDNLA131>3.0.CO;2B.
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)10991506(199807/08)5:4<275::AIDNLA131>3.0.CO;2B
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)10991506(199807/08)5:4<275::AIDNLA131>3.0.CO;2B. 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 GramSchmidt 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 decreasedan increaseonly strategy is described next followed by pseudocode.},
doi = {10.1002/(SICI)10991506(199807/08)5:4<275::AIDNLA131>3.0.CO;2B},
url = {https://www.osti.gov/biblio/433394},
journal = {},
issn = {10705325},
number = 4,
volume = 5,
place = {United States},
year = {1996},
month = {12}
}