Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems

SIAM Journal on Matrix Analysis and Applications

Volumn 29, Issue 3, 2007, Pages 752-773

  Schöberl, Joachim ^a,b   Zulehner, Walter ^c  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

^c JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Conjugate gradient methods; Indefinite preconditioners; KKT systems; Optimal control problems; PDE constrained optimization problems; Saddle point problems

Indexed keywords

CONSTRAINED OPTIMIZATION; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; DISTRIBUTED PARAMETER CONTROL SYSTEMS; EQUATIONS OF STATE; GRADIENT METHODS; LAGRANGE MULTIPLIERS; LEAD; LINEAR SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

APPLIED MATHEMATICS; AUGMENTED LAGRANGIAN APPROACH; CONJUGATE GRADIENT (CG); CONTROL PROBLEMS; CONVERGENCE RATES; COST FUNCTIONAL; DISTRIBUTED CONTROLS; LARGE SCALES; MESH SIZES; NUMERICAL EXPERIMENTS; OPTIMAL CONTROL PROBLEM (OCP); PDE-CONSTRAINED OPTIMIZATION; POSITIVE DEFINITE; PRECONDITIONED CONJUGATE GRADIENT (PCG) METHODS; PRECONDITIONER; PRECONDITIONERS; REGULARIZATION PARAMETERS; SADDLE POINT PROBLEMS; SADDLE POINTS; SCALAR PRODUCTS; SOLUTION METHODS; SPARSE LINEAR SYSTEMS; STATE EQUATIONS;

CONJUGATE GRADIENT METHOD;

EID: 48249114388     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060660977     Document Type: Article

References (27)

1. G. AL-JEIROUDI, J. GONDZIO, AND J. HALL, Preconditioning Indefinite Systems in Interior Point Methods for Large Scale Linear Optimization, Technical report MS-2006-003, School of Mathematics, The University of Edinburgh, Edinburgh, Scotland, 2006.
2. K. ARROW, L. HURWICZ, AND H. UZAWA, Studies in Nonlinear Programming, Stanford University Press, Stanford, CA, 1958.
3. R. E. BANK, B. D. WELFERT, AND H. YSERENTANT, A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1990), pp. 645-666.
4. M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1-137.
5. M. BENZI AND V. SIMONCINI, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103 (2006), pp. 173-196.
6. 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.
7. J. H. BRAMBLE, J. E. PASCIAK, AND A. T. VASSILEV, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072-1092.
8. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
9. H. S. DOLLAR, Iterative Linear Algebra for Constrained Optimization, Ph.D. Thesis, University of Oxford, Oxford, UK, 2005.
10. 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.
11. N. DYN AND W. E. FERGUSON, The numerical solution of equality constrained quadratic programming problems, Math. Comp., 41 (1983), pp. 165-170.
12. H. C. ELMAN AND G. H. GOLUB, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645-1661.
13. B. FISCHER, A. RAMAGE, D. J. SILVESTER, AND A. J. WATHEN, Minimum residual methods for augmented systems, BIT, 38 (1998), pp. 527-543.
14. R. FLETCHER, Practical Methods of Optimization. Vol. 2: Constrained Optimization, John Wiley & Sons, Chichester, UK, 1981.
15. M. FORTIN AND R. GLOWINSKI, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems, North-Holland, Amsterdam, 1983.
16. G. H. GOLUB AND C. GREIF, On solving block-structured indefinite linear systems, SIAM J. Sei. Comput., 24 (2003), pp. 2076-2092.
17. G. H. GOLUB, C. GREIF, AND J. M. VARAH, An algebraic analysis of a block diagonal preconditioner for saddle point problems, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 779-792.
18. N. I. M. GOULD, M. E. HRIBAR, AND J. NOCEDAL, On the solution of equality constrained quadratic programming arising in optimization, SIAM J. Sci. Comput., 23 (2001), pp. 1376-1395.
19. W. HACKBUSCH, Iterative Solutions of Large Sparse Systems of Equations, Springer-Verlag, New York, 1994.
20. 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.
21. 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.
22. T. RUSTEN AND R. WINTHER, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887-904.
23. Y. SAAD AND H. A. VAN DER VORST, Iterative solution of linear systems in the 20th century, J. Comput. Appl. Math., 123 (2000), pp. 1-33.
24. D. SILVESTER AND A. WATHEN, Fast iterative solution of stabilized Stokes systems. Part II: Using block diagonal preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352-1367.
25. F. TRÖLTZSCH, Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen, Vieweg, Wiesbaden, Germany, 2005.
26. P. S. VASSILEVSKI AND R. D. LAZAROV, Preconditioning mixed finite dement saddle-point elliptic problems, Numer. Linear Algebra Appl., 3 (1996), pp. 1-20.
27. W. ZULEHNER, Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp., 71 (2002), pp. 479-505.