Constraint-style preconditioners for regularized saddle point problems

SIAM Journal on Matrix Analysis and Applications

Volumn 29, Issue 2, 2007, Pages 672-684

  Dollar, H S ^a  

^a RUTHERFORD APPLETON LABORATORY   (United Kingdom)

Author keywords

Indefinite linear systems; Krylov subspace methods; Preconditioning

Indexed keywords

CONSTRAINT THEORY; EIGENVALUES AND EIGENFUNCTIONS; LINEAR SYSTEMS; NUMERICAL METHODS;

INDEFINITE LINEAR SYSTEMS; KRYLOV SUBSPACE METHODS; PRECONDITIONING;

PROBLEM SOLVING;

EID: 42649118797     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050626168     Document Type: Article

References (27)

1. A. ALTMAN AND J. GONDZIO, Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization, Optim. Methods Softw., 11/12 (1999), pp. 275-302.
2. O. AXELSSON AND V. A. BARKER, Finite Element Solution of Boundary Value Problems: Theory and Computation, Classics Appl. Math. 35, SIAM, Philadelphia, 2001. Reprint of the 1984 original.
3. O. AXELSSON AND M. NEYTCHEVA, Preconditioning methods for linear systems arising in constrained optimization problems, Numer. Linear Algebra Appl., 10 (2003), pp. 3-31.
4. M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137.
5. L. BERGAMASCHI, J. GONDZIO, M. VENTURIN, AND G. ZILLI, Inexact constraint preconditioners for linear systems arising in interior point methods, Comput. Optim. Appl., 36 (2007), pp. 137-147.
6. G. BIROS AND O. GHATTAS, A Lagrange-Newton-Krylov- Schur method for PDE-constrained optimization, SIAG/Optimization Views-and-News, 11 (2000), pp. 12-18.
7. H. S. DOLLAR, N. I. M. GOULD, W. H. A. SCHILDERS, AND A. J. WATHEN, Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 170-189.
8. C. DURAZZI AND V. RUGGIERO, Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems, Numer. Linear Algebra Appl., 10 (2003), pp. 673-688.
9. H. C. ELMAN, D. J. SILVESTER, AND A. J. WATHEN, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, UK, 2005.
10. R. E. EWING, R. D. LAZAROV, P. LU, AND P. S. VASSILEVSKI, Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic problems, in Preconditioned Conjugate Gradient Methods (Nijmegen, 1989), Lecture Notes in Math. 1457, Springer, Berlin, 1990, pp. 28-43.
11. N. I. M. GOULD, Iterative methods for ill-conditioned linear systems from optimization, in Nonlinear Optimization and Related Topics, G. DiPillo and F. Giannessi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 123-142.
12. N. I. M. GOULD, M. E. HRIBAR, AND J. NOCEDAL, On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23 (2001), pp. 1376-1395.
13. N. I. M. GOULD, D. ORBAN, AND P. L. TOINT, CUTEr and SifDec: a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Software, 29 (2003), pp. 373-394.
14. C. KELLER, N. I. M. GOULD, AND A. J. WATHEN, Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300-1317.
15. A. KLAWONN, An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 540-552.
16. L. LUKŠAN AND J. VLČEK, Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems, Numer. Linear Algebra Appl., 5 (1998), pp. 219-247.
17. L. LUKŠAN AND J. VLČEK, Interior-point method for non-linear non-convex optimization, Numer. Linear Algebra Appl. 11 (2004), pp. 431-453.
18. I. PERUGIA AND V. SIMONCINI, Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl., 7 (2000), pp. 585-616.
19. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994.
20. Y. SAAD, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
21. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869.
22. C. SIEFERT AND E. DE STURLER, Preconditioners for generalized saddle-point problems, SIAM J. Numer. Anal., 44 (2006), pp. 1275-1296.
23. D. SILVESTER AND A. WATHEN, Fast iterative solution of stabilised Stokes systems Part II. Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352-1367.
24. K.-C. TOH, K.-K. PHOON, AND S.-H. CHAN, Block preconditioners for symmetric indefinite linear systems, Internal. J. Numer. Methods Engrg., 60 (2004), pp. 1361-1381.
25. H. A. VAN DER VORST, Iterative Krylov Methods for Large Linear Systems, Cambridge Monogr. Appl. Comput. Math. 13, Cambridge University Press, Cambridge, UK, 2003.
26. R. J. VANDERBEI, Symmetric quasidefinite matrices, SIAM J. Optim., 5 (1995), pp. 100-113.
27. S. J. WRIGHT, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1997.