Symmetric part preconditioning for the conjugate gradient method in Hilbert space. (English) Zbl 1054.65055
The conjugate gradient method for nonsymmetric linear operators in Hilbert space is investigated. The focus is on preconditioning by the symmetric part of the operator, in which case estimates are given for the resulting condition number. An important motivation for this study is given by differential operators, for which the obtained estimates yield mesh independent conditioning properties of the full conjugate gradient method, and are in fact achieved by the simpler truncated version. As an application of the general theory, the conjugate gradient method is used for certain linear elliptic operators on the continuous level in Sobolev spaces.
Reviewer: Karel Najzar (Praha)
MSC:
| 65J10 | Numerical solutions to equations with linear operators |
| 47A50 | Equations and inequalities involving linear operators, with vector unknowns |
Keywords:
Hilbert space; conjugate gradient method; nonsymmetric linear operator; preconditioning; condition number; linear elliptic operatorsReferences:
| [1] | Adams R. A., Sobolev Spaces (1975) |
| [2] | Axelsson O., Iterative Solution Methods (1994) · Zbl 0795.65014 · doi:10.1017/CBO9780511624100 |
| [3] | DOI: 10.1016/0024-3795(80)90226-8 · Zbl 0439.65020 · doi:10.1016/0024-3795(80)90226-8 |
| [4] | DOI: 10.1007/BF01396750 · Zbl 0596.65014 · doi:10.1007/BF01396750 |
| [5] | DOI: 10.1137/0714064 · Zbl 0382.65052 · doi:10.1137/0714064 |
| [6] | Concus P., Lect. Notes Math.Syst. 134 pp 56– (1976) |
| [7] | DOI: 10.1137/0723004 · Zbl 0619.65093 · doi:10.1137/0723004 |
| [8] | DOI: 10.1137/0721026 · Zbl 0546.65010 · doi:10.1137/0721026 |
| [9] | Hackbusch W., Theorie und Numerik Elliptischer Differentialgleichungen (1986) · Zbl 0609.65065 · doi:10.1007/978-3-322-99946-7 |
| [10] | Hanke M., Pitman Research Notes,Math. Ser. Longman pp 327– (1995) |
| [11] | Hayes R. M., Nat. Bur. Standards Appl. Math. Ser. 39 pp 71– (1954) |
| [12] | DOI: 10.1137/0709016 · Zbl 0243.65026 · doi:10.1137/0709016 |
| [13] | Lions J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969) |
| [14] | DOI: 10.1137/0730040 · Zbl 0782.65048 · doi:10.1137/0730040 |
| [15] | Riesz F., Vorlesungen über Funktionalanalysis (1982) |
| [16] | DOI: 10.1090/S0025-5718-1985-0777273-9 · doi:10.1090/S0025-5718-1985-0777273-9 |
| [17] | DOI: 10.1137/0907058 · Zbl 0599.65018 · doi:10.1137/0907058 |
| [18] | Simon L., Linear Partial Differential Equations of Second Order (1983) |
| [19] | DOI: 10.1137/1019071 · Zbl 0358.65088 · doi:10.1137/1019071 |
| [20] | Voevodin V. V., Math. Math. Phys. 23 pp 143– (1983) · Zbl 0547.65032 · doi:10.1016/S0041-5553(83)80060-3 |
| [21] | DOI: 10.1137/0715053 · Zbl 0398.65030 · doi:10.1137/0715053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
