A comparison of deflation and the balancing preconditioner

SIAM Journal on Scientific Computing

Volumn 27, Issue 5, 2006, Pages 1742-1759

  Nabben, R ^a   Vuik, C ^b  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^b DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Balancing; Coarse grid correction; Conjugate gradients; Deflation; Porous media flow; Preconditioners; Scalable parallel preconditioner

Indexed keywords

BALANCING; COMPUTATIONAL METHODS; DECOMPOSITION; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; NUMERICAL METHODS; POROUS MATERIALS; VECTORS;

COARSE GRID CORRECTION; CONJUGATE GRADIENTS; DEFLATION; POROUS MEDIA FLOW; PRECONDITIONERS; SCALABLE PARALLEL PRECONDITIONER;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33748764169     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040608246     Document Type: Conference Paper

References (26)

1. J. H. BRAMBLE, J. E. PASCIAK, AND A. H. SCHATZ, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp., 47 (1986), pp. 103-134.
2. H. DE GERSEM AND K. HAMEYER, A deflated iterative solver for magnetostatic finite element models with large differences in permeability, Eur. Phys. J. Appl. Phys., 13 (2000), pp. 45-49.
3. M. DRYJA AND O. B. WIDLUND, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-155.
4. M. EIERMANN, O. G. ERNST, AND O. SCHNEIDER, Analysis of acceleration strategies for restarted minimal residual methods, J. Comput. Appl. Math., 123 (2000), pp. 261-292.
5. C. FARHAT AND F.-X. ROUX, A method of finite element tearing and interconnecting and its parallel solution algorithm, Internat. J. Numer. Methods Engrg., 32 (1991), pp. 1205-1227.
6. C. FARHAT AND F.-X. ROUX, Implicit parallel processing in structural mechanics, Comput. Mech. Adv., 2 (1994), pp. 1-124.
7. J. FRANK AND C. VUIK, On the construction of deflation-based preconditioners, SIAM J. Sci. Comput., 23 (2001), pp. 442-462.
8. R. A. HORN AND C. R. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
9. E. F. KAASSCHIETER, Preconditioned conjugate gradients for solving singular systems, J. Comput. Appl. Math., 24 (1988), pp. 265-275.
10. L. YU. KOLOTILINA, Preconditioning of systems of linear algebraic equations by means of twofold deflation. I. Theory, J. Math. Sci. (New York), 89 (1998), pp. 1652-1689.
11. J. MANDEL, Balancing domain decomposition, Comm. Numer. Methods Engrg., 9 (1993), pp. 233-241.
12. J. MANDEL AND M. BREZINA, Balancing domain decomposition for problems with large jumps in coefficients, Math. Comp., 216 (1996), pp. 1387-1401.
13. L. MANSFIELD, Damped Jacobi preconditioning and coarse grid deflation for conjugate gradient iteration on parallel computers, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 1314-1323.
14. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
15. R. B. MORGAN, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1154-1171.
16. R. NABBEN AND C. VUIK, A comparison of deflation and coarse grid correction applied to porous media flow, SIAM J. Numer. Anal., 42 (2004), pp. 1631-1647.
17. R. A. NICOLAIDES, Deflation of conjugate gradients with applications to boundary value problems, SIAM J. Numer. Anal., 24 (1987), pp. 355-365.
18. L. F. PAVARINO AND O. B. WIDLUND, Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math., 55 (2002), pp. 302-335.
19. A. QUARTERONI AND A. VALLI, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, Oxford, UK, 1999.
20. B. SMITH, P. BJØRSTAD, AND W. GROPP, Domain Decomposition, Cambridge University Press, Cambridge, UK, 1996.
21. A. TOSELLI AND W. WIDLUND, Domain Decomposition Methods, Springer Ser. Comput. Math. 34, Springer-Verlag, Berlin, 2005.
22. A. VAN DER SLUIS AND H. A. VAN DER VORST, The rate of convergence of conjugate gradients, Numer. Math., 48 (1986), pp. 543-560.
23. F. VERMOLEN, C. VUIK, AND A. SEGAL, Deflation in preconditioned conjugate gradient methods for finite element problems, in Conjugate Gradient and Finite Element Methods, M. Krizek, P. Neittaanmäki, R. Glowinski, and S. Korotov, eds., Springer-Verlag, Berlin, 2004, pp. 103-129.
24. C. VUIK, A. SEGAL, J. A. MEIJERINK, AND G. T. WIJMA, The construction of projection vectors for a deflated ICCG method applied to problems with extreme contrasts in the coefficients, J. Comput. Phys., 172 (2001), pp. 426-450.
25. C. VUIK, A. SEGAL, AND J. A. MEIJERINK, An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients, J. Comput. Phys., 152 (1999), pp. 385-403.
26. C. VUIK, A. SEGAL, L. EL YAAKOUBI, AND E. DUFOUR, A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients, Appl. Numer. Math., 41 (2002), pp. 219-233.