Abstract
In this paper, I present an efficient reordering-based preconditioner for the elliptic PDE-constrained optimization problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under certain suitable conditions. In addition, the spectral properties of the corresponding preconditioned matrix are studied. The resulting preconditioner is efficient for solving problems with small Tikhonov parameters (less than \(1e{-}6\)). Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner.
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References
Arrow K, Hurwicz L, Uzawa H (1958) Studies in nonlinear programming. Stanford University Press, Stanford
Bai ZZ (2011) Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91:379–395
Bai ZZ, Benzi M, Chen F (2013) Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J Numer Anal 33:343–369
Bramble JH, Pasciak JE, Vassilev AT (1997) Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J Numer Anal 34:1072–1092
Elman HC, Golub GH (1994) Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal 31:1645–1661
Elman HC, Silvester DJ, Wathen AJ (2005) Finite elements and fast iterative solvers: with applications in incompressible fiuid dynamics. Oxford University Press, Oxford
Elman HC, Ramage A, Silvester DJ (2007) Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow. ACM Trans Math Softw 33:1–18
Golub GH, van Loan CF (2013) Matrix computations, 4th edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore
Huang N, Ma CF (2017) Analysis on inexact block diagonal preconditioners for elliptic PDE-constrained optimization problems. Comput Math Appl 74:2423–2437
Ke YF, Ma CF (2018) Some preconditioners for elliptic PDE-constrained optimization problems. Comput Math Appl 75:2795–2813
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
Mirchi H, Salkuyeh DK (2020) A new preconditioner for elliptic PDE-constrained optimization problems. Numer Algorithms 83:653–668
Pearson JW, Wathen AJ (2012) A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer Linear Algebra Appl 19:816–829
Rees T, Dollar HS, Wathen AJ (2010) Optimal solvers for PDE-constrained optimization. SIAM J Sci Comput 32:271–298
Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia
Zeng YP, Wang SQ, Xu HR, Xie SL (2015) Preconditioners for reduced saddle point systems arising in elliptic PDE-constrained optimization problems. J Inequal Appl 355:1–14
Zheng QQ, Lu LZ (2017) A shift-splitting preconditioner for a class of block two-by-two linear systems. Appl Math Lett 66:54–60
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The author would like to thank the reviewers for their detailed comments and suggestions that led to considerable improvements in the quality of the present manuscript.
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Communicated by Xiang Wang.
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The author was supported by National Natural Science Foundation of China Grant (No. 12001311), Science Foundation of China University of Petroleum, Beijing (No. 2462021YJRC025), and the State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum
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Zheng, Q. A reordering-based preconditioner for elliptic PDE-constrained optimization problems with small Tikhonov parameters. Comp. Appl. Math. 42, 169 (2023). https://doi.org/10.1007/s40314-023-02317-7
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DOI: https://doi.org/10.1007/s40314-023-02317-7