Abstract
For the structured systems of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems, we construct block-counter-diagonal and block-counter-tridiagonal preconditioning matrices to precondition the Krylov subspace methods such as GMRES. We derive explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices. Numerical implementations show that these structured preconditioners may lead to satisfactory experimental results of the preconditioned GMRES methods when the regularization parameter is suitably small.
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Bai Z-Z (2004) Construction and analysis of structured preconditioners for block two-by-two matrices. J Shanghai Univ (English Ed) 8: 397–405
Bai Z-Z (2006) Structured preconditioners for nonsingular matrices of block two-by-two structures. Math Comput 75: 791–815
Bai Z-Z, Ng MK (2005) On inexact preconditioners for nonsymmetric matrices. SIAM J Sci Comput 26: 1710–1724
Bai Z-Z, Ng MK, Wang Z-Q (2009) Constraint preconditioners for symmetric indefinite matrices. SIAM J Matrix Anal Appl 31: 410–433
Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14: 1–137
Elman HC, Silvester DJ, Wathen AJ (2005) Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, Oxford/New York
Keller C, Gould NIM, Wathen AJ (2000) Constraint preconditioning for indefinite linear systems. SIAM J Matrix Anal Appl 21: 1300–1317
Lass O, Vallejos M, Borzi A, Douglas CC (2009) Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84: 27–48
Lions JL (1968) Optimal Control of Systems. Springer, Berlin/Heidelberg
Murphy MF, Golub GH, Wathen AJ (2000) A note on preconditioning for indefinite linear systems. SIAM J Sci Comput 21: 1969–1972
Pan J-Y, Ng MK, Bai Z-Z (2006) New preconditioners for saddle point problems. Appl Math Comput 172: 762–771
Rees T, Dollar HS, Wathen AJ (2008) Optimal solvers for PDE-constrained optimization, Technical Report of Oxford University Computing Laboratory, No. 08/10
Silvester DJ, Wathen AJ (1994) Fast iterative solution of stabilised Stokes systems. Part II: using general block preconditioners. SIAM J Numer Anal 31: 1352–1367
Wathen AJ (1987) Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J Numer Anal 7: 449–457
Wathen AJ, Silvester DJ (1993) Fast iterative solution of stabilised Stokes systems. Part I: using simple diagonal preconditioners. SIAM J Numer Anal 30: 630–649
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Communicated by C.C. Douglas.
This study was supported by The National Basic Research Program (No. 2005CB321702) and The National Outstanding Young Scientist Foundation (No. 10525102), P.R. China.
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Bai, ZZ. Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011). https://doi.org/10.1007/s00607-010-0125-9
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DOI: https://doi.org/10.1007/s00607-010-0125-9