Preconditioning saddle-point systems with applications in optimization

SIAM Journal on Scientific Computing

Volumn 32, Issue 1, 2010, Pages 249-270

  Dollar, H Sue ^a   Gould, Nicholas I M ^a   Stoll, Martin ^b   Wathen, Andrew J ^b  

^a RUTHERFORD APPLETON LABORATORY   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Iterative solver; Optimization; Preconditioning; Saddle point problems

Indexed keywords

CONJUGATE GRADIENT METHOD;

APPLICATION AREA; EXTREMUM PRINCIPLES; INCOMPRESSIBLE FLUID; ITERATIVE SOLVERS; NEW APPROACHES; OPTIMISATIONS; PRECONDITIONING; SADDLE POINT PROBLEMS; SADDLE POINT SYSTEM; STOKES PROBLEM;

MACHINERY;

EID: 77249106506     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080727129     Document Type: Conference Paper

References (36)

1. S. F. ASHBY, T. A. MANTEUFFEL, AND P. E. SAYLOR, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542-1568.
2. S. F. ASHBY, M. J. HOLST, T. A. MANTEUFFEL, AND P. E. SAYLOR, The role of the inner product in stopping criteria for conjugate gradient iterations, BIT, 41 (2001), pp. 26-52.
3. M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137.
4. M. BENZI AND V. SIMONCINI, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103 (2006), pp. 173-196.
5. J. H. BRAMBLE AND J. E. PASCIAK, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp., 50 (1988), pp. 1-17.
6. S. H. CHENG AND N. J. HIGHAM, A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 1097-1110.
7. H. C. ELMAN, A. RAMAGE, AND D. J. SILVESTER, Algorithm 886: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Software, 33 (2007), Art. 14.
8. H. C. ELMAN, D. J. SILVESTER, AND A. J. WATHEN, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Numer. Math. Sci. Comput., Oxford University Press, New York, 2005.
9. H. FANG AND D. O'LEARY, Modified Cholesky algorithms: A catalog with new approaches, Math. Program. Ser. A, 115 (2008), pp. 319-349.
10. B. FISCHER, A. RAMAGE, D. J. SILVESTER, AND A. J. WATHEN, Minimum residual methods for augmented systems, BIT, 38 (1998), pp. 527-543.
11. A. FORSGREN, P.E.GILL, AND J. D. GRIFFIN, Iterative solution of augmented systems arising in interior methods, SIAM J. Optim., 18 (2007), pp. 666-690.
12. R. W. FREUND, Transpose-free quasi-minimal residual methods for non-Hermitian linear systems, in Recent Advances in Iterative Methods, IMA Vol. Math. Appl. 60, Springer, New York, 1994, pp. 69-94.
13. P. E. GILL AND W. MURRAY, Newton-type methods for unconstrained and linearly constrained optimization, Math. Programming, 7 (1974), pp. 311-350.
14. P. E. GILL, W. MURRAY, AND M. H. WRIGHT, Practical Optimization, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London-New York, 1981.
15. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins Stud. Math. Sci., Johns Hopkins University Press, Baltimore, MD, 1996.
16. N. GOULD, D. ORBAN, AND P. TOINT, CUTEr and SifDec: A constrained and unconstrained testing environment, revisited, ACM Trans. Math. Software, 29 (2003), pp. 373-394.
17. N. GOULD, D. ORBAN, AND P. L. TOINT, CUTEr, a Constrained and Unconstrained Testing Environment, Revisited, Technical report TR/PA/01/04, CERFACS, Toulouse, France, 2001.
18. N. GOULD AND V. SIMONCINI, Spectral Analysis of Saddle Point Matrices with Indefinite Leading Blocks, Technical report, Rutherford Appleton Laboratory, Chilton, England, 2008.
19. N. I. M. GOULD, On the accurate determination of search directions for simple differentiable penalty functions, IMA J. Numer. Anal., 6 (1986), pp. 357-372.
20. M. H. GUTKNECHT, Changing the norm in conjugate gradient type algorithms,SIAM J. Numer. Anal., 30 (1993), pp. 40-56.
21. 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.
22. A. KLAWONN, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 172-184. (Pubitemid 128617506)
23. J. LIESEN AND B. N. PARLETT, On nonsymmetric saddle point matrices that allow conjugate gradient iterations, Numer. Math., 108 (2008), pp. 605-624.
24. A. MEYER AND T. STEIDTEN, Improvements and Experiments on the Bramble-Pasciak Type CG for Mixed Problems in Elasticity, Technical report SFB393/01-13, TU Chemnitz, Chemnitz, Germany, 2001.
25. M. F. MURPHY, G. H. GOLUB, AND A. J. WATHEN, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969-1972.
26. C. C. PAIGE AND M. A. SAUNDERS, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
27. M. ROZLOŽNÍK AND V. SIMONCINI, Krylov subspace methods for saddle point problems with indefinite preconditioning, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 368-391.
28. Y. SAAD, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
29. M. A. SAUNDERS, Cholesky-based methods for sparse least squares: The benefits of regularization, in Linear and Nonlinear Conjugate Gradient-Related Methods (Seattle, WA, 1995), Loyce Adams and J. L. Lazareth, eds., SIAM, Philadelphia, 1996, pp. 92-100.
30. R. B. SCHNABEL AND E. ESKOW, A new modified Cholesky factorization, SIAM J.Sci. Statist. Comput., 11 (1990), pp. 1136-1158.
31. R. B. SCHNABEL AND E. ESKOW, A revised modified Cholesky factorization algorithm, SIAM J. Optim., 9 (1999), pp. 1135-1148.
32. J. SCHÖBERL AND W. ZULEHNER, Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 752-773.
33. V. SIMONCINI, Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math., 49 (2004), pp. 63-80.
34. M. STOLL, Solving Linear Systems Using the Adjoint, Ph.D. thesis, University of Oxford, Oxford, UK, 2009.
35. M. STOLL AND A. WATHEN, Combination preconditioning and the Bramble-Pasciak+ preconditioned SIAM J. Matrix Anal. Appl., 30 (2008), pp. 582-608.
36. S. J. WRIGHT, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1997.