Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 1, 2006, Pages 170-189

  Dollar, H Sue ^a   Gould, Nicholas I M ^b   Schilders, Wil H A ^c,d   Wathen, Andrew J ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b RUTHERFORD APPLETON LABORATORY   (United Kingdom)

^c PHILIPS RESEARCH LABORATORIES   (Netherlands)

^d EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Implicit factorization preconditioners; Regularized saddle point systems

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; NUMERICAL ANALYSIS; PROBLEM SOLVING;

CLUSTERING; IMPLICIT-FACTORIZATION PRECONDITIONERS; SADDLE-POINT SYSTEMS;

ITERATIVE METHODS;

EID: 33847092908     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/05063427X     Document Type: Article

References (49)

1. O. AXELSSON AND M. NEYTCHEVA, Preconditioning methods for linear systems arising in constrained optimization problems, Numer. Linear Algebra Appl., 10 (2003), pp. 3-31.
2. R. H. BARTELS AND G. H. GOLUB, The simplex method of linear programming using lu decompositions, Comm. of the ACM, 12 (1969), pp. 266-268.
3. M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137.
4. L. BERGAMASCHI, J. GONDZIO, AND G. ZILLI, Preconditioning indefinite systems in interior point methods for optimization, Comput. Optim. Appl., 28 (2004), pp. 149-171.
5. G. BIROS AND O. GHATTAS, A Lagrange-Newton-Krylov- Schur method for PDE-constrained optimization, SIAG/OPT Views-and-News, 11 (2000), pp. 12-18.
6. J. H. BRAMBLE AND J. E. PASCIAK, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comput., 50 (1988), pp. 1-17.
7. R. H. BYRD, M. E. HRIBAR, AND J. NOCEDAL, An interior point algorithm for large scale nonlinear programming, SIAM J. Optim., 9 (2000), pp. 877-900.
8. T. F. COLEMAN, Linearly constrained optimization and projected preconditioned conjugate gradients, in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, J. Lewis, ed., SIAM Philadelphia, 1994, pp. 118-122.
9. T. F. COLEMAN AND A. POTHEN, The null space problem I: Complexity, SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 527-537.
10. T. F. COLEMAN AND A. POTHEN, The null space problem II: Algorithms, SIAM J. Algebraic Discrete Methods, 8 (1987), pp. 544-563.
11. T. F. COLEMAN AND A. VERMA, A preconditioned conjugate gradient approach to linear equality constrained minimization, Technical report, Department of Computer Sciences, Cornell University, Ithaca, New York, 1998.
12. P. CONCUS, G. H. GOLUB, AND D. P. O'LEARY, Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method, in Sparse Matrix Computations, J. Bunch and D. Rose, eds., Academic Press, London, 1976, pp. 309-332,
13. A. R. CONN, N. I. M. GOULD, D. ORBAN, AND PH. L. TOINT, A primal-dual trust-region algorithm for non-convex nonlinear programming, Math. Prog., 87 (2000), pp. 215-249.
14. E. DE STURLER AND J. LIESEN, Block-diagonal and constraint preconditioned for nonsymmetric indefinite linear systems Part I; Theory, SIAM J. Sci. Comput., 26 (2005), pp. 1598-1619.
15. E. D. DOLAN AND J. J. MORÉ, Benchmarking optimization software with performance profiles, Math. Prog., 91 (2002), pp. 201-213.
16. H. S. DOLLAR, Extending constraint preconditioned for saddle-point problems, Technical report NA-05-02, Oxford University Computing Laboratory, Oxford, England, 2005. (Submitted to SIAM J. Matrix Anal. Appl.)
17. H. S. DOLLAR, N. I. M. GOULD, W. H. A. SCHILDERS, AND A. J. WATHEN, On iterative methods and implicit-factorization preconditioned for regularized saddle-point systems, Technical report RAL-TR-2005-011, Rutherford Appleton Laboratory, Chilton, Oxfordshire, England, 2005.
18. H. S. DOLLAR, N. I. M. GOULD, AND A. J. WATHEN, On implicit-factorization constraint preconditioned, Nonconvex Optimization and Its Applications 83, Springer Verlag, 2006.
19. H. S. DOLLAR AND A. J. WATHEN, Incomplete factorization constraint preconditioned for saddle point problems, Technical report 04-01, Oxford University Computing Laboratory, Oxford, England, 2004. (To appear in SIAM J. Sci. Comput.)
20. I. S. DUFF, MA57 - a code for the solution of sparse symmetric definite and indefinite systems, ACM Trans. Math. Software, 30 (2004), pp. 118-144.
21. I. S. DUFF AND J. K. REID, The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software, 9 (1983), pp. 302-325.
22. C. DURAZZI AND V. RUOGIERO, Indefinitely constrained conjugate gradient method for large sparse equality and inequality constrained quadratic problems, Numer. Linear Algebra Appl., 10 (2002), pp. 673-688.
23. H. C. ELMAN, D. J. SILVESTER, AND A. J. WATHEN, Finite-Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, London, 2005.
24. 3. J. H. FORREST AND J, A. TOMLIN, Updating triangular factors of the basis to maintain sparsity in the product form simplex method, Math. Prog., 2 (1972), pp. 263-278.
25. R. W. FREUND AND N. M. NACHTIGAL, A new Krylov-subspace method for symmetric indefinite linear systems, in Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, W. F. Ames, ed., IMACS, 1994, pp. 1253-1256.
26. P. E. GILL, W. MURRAY, D. B. PONCELEÓN, AND M. A. SAUNDERS, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292-311.
27. P. E. GILL, W. MURRAY, AND M. A. SAUNDERS, SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM J. Optim., 12 (2002), pp. 979-1006.
28. P. E. GILL AND M. A. SAUNDERS, private communication, 1999.
29. N. I. M. GOULD, On practical conditions for the existence and uniqueness of solutions to the general equality quadratic-programming problem, Math. Prog., 32 (1985), pp. 90-99.
30. N. I. M. GOULD, Iterative methods for ill-conditioned linear systems from optimization, in Nonlinear Optimization and Related Topics, G. Di Pillo and F. Giannessi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 123-142.
31. N. I. M. GOULD, M. E. HRIBAR, AND J. NOCEDAL, On the solution of equality constrained quadratic problems arising in optimization, SIAM J. Sci. Comput., 23 (2001), pp. 1375-1394.
32. N. I. M. GOULD, D. ORBAN, AND PH. L. TOINT, CUTEr (and SifDec), a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Software, 29 (2003), pp. 373-394.
33. C. GREIF, G. H. GOLUB, AND J. M. VARAH, Augmented Lagrangian techniques for solving saddle point linear systems, Technical report, Computer Science Department, University of British Columbia, Vancouver, Canada, 2004.
34. HSL, A collection of Fortran codes for large-scale scientific computation, see http://hsl.rl.ac.uk, 2004.
35. 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.
36. L. LUKŠAN AND J. VLČEK, Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems, Numer. Linear Algebra Appl., 5 (1998), pp. 219-247.
37. M. MIHAJLOVIC AND D. SILVESTER, A black-box multigrid preconditioner for the biharmonic equation, BIT, 44 (2004), pp. 151-163.
38. B. A. MURTAGH AND M. A. SAUNDERS, Large-scale linearly constrained optimization, Math. Prog., 14 (1978), pp. 41-72.
39. B. A. MURTAGH AND M. A. SAUNDERS, A projected Lagrangian algorithm and its implementation for sparse non-linear constraints, Math. Programming Stud., 16 (1982), pp. 84-117.
40. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer Series in Operations Research, Springer, New York, 1999.
41. L. A. PAVARINO, Preconditioned mixed spectral finite-element methods for elasticity and Stokes problems, SIAM J. Sci. Comput., 19 (1998), pp. 1941-1957.
42. I. PERUGIA AND V. SIMONCINI, Block-diagonal and indefinite symmetric preconditioned for mixed finite element formulations, Numer. Linear Algebra Appl., 7 (2000), pp. 585-616.
43. B. T. POLYAK, The conjugate gradient method in extremal problems, U.S.S.R. Comput. Math. Math. Phys., 9 (1969), pp. 94-112.
44. M. ROZLOZNÍK AND V. SIMONCINI, Krylov subspace methods for saddle point problems with indefinite preconditioning, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 368-391.
45. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput., 7 (1986), pp. 856-869.
46. W. SCHILDERS, A preconditioning technique for indefinite systems arising in electronic circuit simulation, Talk at the one-day meeting on preconditioning methods for indefinite linear systems, Technische Universiteit, Eindhoven, The Netherlands, December 9, 2002.
47. W. H. A. SCHILDERS AND E. J. W. TER MATEN, Numerical methods in electromagnetics, in Handbook of Numerical Analysis Vol XIII, P. G. Ciarlet, ed., Elsevier, North Holland, Amsterdam, 2005.
48. K. TOH, K. PHOON, AND S. CHAN, Block preconditioned for symmetric indefinite linear systems, Internat. J. Numer. Methods Engrg., 60 (2004), pp. 1361-1381.
49. R. J. VANDERBEI AND D. F. SHANNO, An interior point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999), pp. 231-252.