Block-triangular preconditioners for PDE-constrained optimization

Numerical Linear Algebra with Applications

Volumn 17, Issue 6, 2010, Pages 977-996

  Rees, Tyrone ^a   Stoll, Martin ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Krylov subspaces; Linear systems; PDE constrained optimization; Preconditioning; Saddle point problems

Indexed keywords

EID: 78649645171     PISSN: 10705325     EISSN: 10991506     Source Type: Journal    
DOI: 10.1002/nla.693     Document Type: Article

References (39)

1. Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numerica 2005; 14:1-137.
2. Paige CC, Saunders MA. Solutions of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis 1975; 12(4):617-629.
3. Hestenes MR, Stiefel E. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 1952; 49:409-436 (1953).
4. Ashby S, Manteuffel T, Saylor P. A taxonomy for conjugate gradient methods. SIAM Journal on Numerical Analysis 1990; 27(6):1542-1568.
5. Dollar S. Iterative linear algebra for constrained optimization. Ph.D. Thesis, University of Oxford, 2005. Available from:
6. Tröltzsch F. Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg Verlag: Wiesbaden, 2005.
7. Hinze M, Pinnau R, Ulbrich M, Ulbrich S. Mathematical modelling: theory and applications. Optimization with PDE Constraints. Springer: New York, 2009.
8. Stoll M, Wathen A. Preconditioning for active set and projected gradient methods as semi-smooth Newton methods for PDE-constrained optimization with control constraints. SIAM Journal on Optimization, 2009; submitted.
9. Elman HC, Silvester DJ, Wathen AJ. Numerical mathematics and scientific computation. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press: New York, 2005.
10. Rees T, Dollar HS, Wathen A. Optimal solvers for PDE-constrained optimization. SIAM Journal on Scientific Computing 2008; accepted for publication.
11. Bramble JH, Pasciak JE. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation 1988; 50(181):1-17.
12. Wathen A, Silvester D. Fast iterative solution of stabilised Stokes systems. I. Using simple diagonal preconditioners. SIAM Journal on Numerical Analysis 1993; 30(3):630-649.
13. Elman HC. Multigrid and Krylov subspace methods for the discrete Stokes equations. In Seventh Copper Mountain Conference on Multigrid Methods, Melson ND, Manteuffel TA, McCormick SF, Douglas CC (eds), vol. CP 3339. NASA: Hampton, VA, 1996; 283-299. Available from:
14. Klawonn A. Block-triangular preconditioners for saddle point problems with a penalty term. SIAM Journal on Scientific Computing 1998; 19(1):172-184. Special issue on iterative methods (Copper Mountain, CO, 1996).
15. Carstensen C, Kuhn M, Langer U. Fast parallel solvers for symmetric boundary element domain decomposition equations. Numerische Mathematik 1998; 79(3):321-347.
16. Ainsworth M, Sherwin S. Domain decomposition preconditioners for p and hp finite element approximation of Stokes equations. Computer Methods in Applied Mechanics and Engineering 1999; 175(3-4):243-266.
17. Zulehner W. Analysis of iterative methods for saddle point problems: a unified approach. Mathematics of Computation 2002; 71(238):479-505.
18. Gatica GN, Heuer N. Conjugate gradient method for dual-dual mixed formulations. Mathematics of Computation 2002; 71(240):1455-1472.
19. Axelsson O, Neytcheva M. Preconditioning methods for linear systems arising in constrained optimization problems. Numerical Linear Algebra with Applications 2003; 10(1-2):3-31. Dedicated to the 60th birthday of Raytcho Lazarov.
20. Axelsson O, Neytcheva M. Robust preconditioners for saddle point problems. Numerical Methods and Applications. Lecture Notes in Computer Science, vol. 2542. Springer: Berlin, 2003, 158-166.
21. Peters J, Reichelt V, Reusken A. Fast iterative solvers for discrete Stokes equations. SIAM Journal on Scientific Computing 2005; 27(2):646-666.
22. Murphy MF, Golub GH, Wathen AJ. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing 2000; 21(6):1969-1972.
23. Stoll M. Solving linear systems using the adjoint. Ph.D. Thesis, University of Oxford, 2009.
24. Elman HC. Preconditioners for saddle point problems arising in computational fluid dynamics. Applied Numerical Mathematics 2002; 43(1-2):75-89. 19th Dundee Biennial Conference on Numerical Analysis, 2001.
25. + preconditioner. SIAM Journal on Matrix Analysis and Applications 2008; 30(2):582-608. DOI: 10.1137/070688961. Available from:
26. Simoncini V. Block triangular preconditioners for symmetric saddle-point problems. Applied Numerical Mathematics 2004; 49(1):63-80.
27. Rozlozník M, Simoncini V. Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM Journal on Matrix Analysis and Applications 2002; 24(2):368-391. DOI: 10.1137/S0895479800375540. URL:
28. Saad Y. Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia, PA, 2003.
29. Rees T, Stoll M, Wathen A. All-at-once preconditioners for PDE-constrained optimization. Kybernetica 2009; submitted.
30. Boyle J, Mihajlovic MD, Scott JA. HSL_MI20: an efficient AMG preconditioner. Technical Report RAL-TR-2007-021, Rutherford Appleton Laboratory (CCLRC), 2007.
31. Wathen A, Rees T. Chebyshev semi-iteration in preconditioning. Technical Report NA-08/14, Oxford University Computing Laboratory, 2008.
32. Wathen AJ. Realistic eigenvalue bounds for the Galerkin mass matrix. IMA Journal of Numerical Analysis 1987; 7(4):449-457.
33. Wathen AJ. On relaxation of Jacobi iteration for consistent and generalized mass matrices. Communications in Applied Numerical Methods 1991; 7(2):93-102.
34. Varga RS. Matrix Iterative Analysis. Prentice-Hall Inc.: Englewood Cliffs, NJ, 1962.
35. Hageman LA, Young DM. Applied Iterative Methods. Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: New York, 1981. Computer Science and Applied Mathematics.
36. Golub GH, Varga RS. Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. I. Numerical Mathematics 1961; 3:147-156.
37. Golub GH, Varga RS. Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods. II. Numerical Mathematics 1961; 3:157-168.
38. Braess D, Peisker P. On the numerical solution of the biharmonic equation and the role of squaring matrices for preconditioning. IMA Journal of Numerical Analysis 1986; 6(4):393-404.
39. Silvester D, Wathen A. Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners. SIAM Journal on Numerical Analysis 1994; 31(5):1352-1367.