Skip to main content
SpringerLink
Log in

PMHSS iteration method and preconditioners for Stokes control PDE-constrained optimization problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique can achieve satisfactory results for solving optimal control problems governed by Poisson’s equation. We explore the feasibility of such a method and preconditioner for solving optimization problems constrained by the more complicated Stokes system. Theoretical results demonstrate that the PMHSS iteration method is convergent because the spectral radius of the iterative matrix is less than \(\frac {\sqrt {2}}{2}\). Additionally, the PMHSS preconditioner still clusters eigenvalues on a unitary segment. It guarantees that the convergence of the PMHSS iteration method and preconditioning is independent of not only discretizing mesh size, but also of the Tikhonov regularization parameter. A more effective preconditioner is proposed based on the PMHSS preconditioner. The proposed preconditioner avoids the inner iterations when solving saddle point systems appearing in the generalized residual equations. Furthermore, it is still convergent and maintains its independence of parameter and mesh size.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems Stokes control. Numer. Algorithms 74, 19–37 (2017)

    Article  MathSciNet  Google Scholar 

  2. Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66(4), 811–841 (2014)

    Article  MathSciNet  Google Scholar 

  3. Axelsson, O., Davod, S.K.: A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Numer. Math. 59(2), 321–342 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: 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(1), 343–369 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93(1), 41–60 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numerische Mathematik 102 (1), 1–38 (2005)

    Article  MathSciNet  Google Scholar 

  7. Bai, Z.-Z., Wang, Z.-Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra and its Applications 428, 2900–2932 (2008)

    Article  MathSciNet  Google Scholar 

  8. Boyle, J., Mihajlovic, M.D., Scott, J.A.: HSL_MI20 An Efficient AMG preconditioner for finite element problems in 3D. Int. J. Numer. Methods Eng. 82(1), 64–98 (2009)

    Article  Google Scholar 

  9. Cahouet, J., Chabard, J.P.: Some finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8(8), 869–895 (1988)

    Article  MathSciNet  Google Scholar 

  10. Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994)

    Article  MathSciNet  Google Scholar 

  11. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, USA (2014)

    Book  Google Scholar 

  12. Jiang, E.: Bounds for the smallest singular value of a Jordan block with an application to eigenvalue perturbation. Linear Algebra and its Applications 197, 691–707 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Liao, L., Zhang, G.-F.: The generalized C-to-R method for solving complex symmetric indefinite linear systems. Linear and Multilinear Algebra 67, 1–9 (2019)

    Article  MathSciNet  Google Scholar 

  14. Pearson, J.W.: On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems. Electornic Trans. Numer. Anal. 44, 53–72 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Rees, T., Sue Dollar, H., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32(1), 271–298 (2010)

    Article  MathSciNet  Google Scholar 

  16. Ren, Z.-R., Cao, Y., Zhang, L.-L.: On preconditioned MHSS real-valued iteration methods for a class of complex symmetric indefinite linear systems. East Asian J. Appl. Math. 6(2), 192–210 (2016)

    Article  MathSciNet  Google Scholar 

  17. Shen, Q.-Q., Shi, Q.: A variant of the HSS preconditioner for complex symmetric indefinite linear systems. Computers & Mathematics with Applications 75(3), 850–863 (2018)

    Article  MathSciNet  Google Scholar 

  18. Wang, Zeng-Qi: Restrictively preconditioned Chebyshev method for solving systems of linear equations. J. Eng. Math. 93(1), 61–76 (2015)

    Article  MathSciNet  Google Scholar 

  19. Wang, Z.-Q.: On a Chebyshev accelerated splitting iteration method with application to two-by-two block linear systems. Numerical Linear Algebra with Applications 25(5), e2712 (2018)

    Article  MathSciNet  Google Scholar 

  20. Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449–457 (1987)

    Article  MathSciNet  Google Scholar 

  21. Wu, S.-L., Li, C.-X.: Modified complex-symmetric and skew-Hermitian splitting iteration method for a class of complex-symmetric indefinite linear systems. Numer. Algorithms 76(1), 93–107 (2017)

    Article  MathSciNet  Google Scholar 

  22. Xu, W.-: A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems. Appl. Math. Comput. 219(21), 10510–10517 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Zhang, J.-H., Dai, H.: A new block preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms 74(3), 889–903 (2017)

    Article  MathSciNet  Google Scholar 

  24. Zhang, J.-L., Fan, H.-T., Gu, C.-Q.: An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms 77 (2), 451–478 (2018)

    Article  MathSciNet  Google Scholar 

  25. Zulehner, W.: Nonstandard norms and robust estimates for saddle point problems. SIAM Journal on Matrix Analysis and Applications 32(2), 536–560 (2011)

    Article  MathSciNet  Google Scholar 

  26. Zulehner, W.: Efficient solvers for saddle point problems with applications to PDE-constrained optimization, pp 197–216. Springer, Berlin (2013)

    MATH  Google Scholar 

Download references

Funding

This research is supported by the National Natural Science Foundation of China (11371022).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    Shan-Mou Cao

  2. School of Mathematical Sciences and Ministry of Education Key Lab in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, China

    Zeng-Qi Wang

Authors
  1. Shan-Mou Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeng-Qi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zeng-Qi Wang.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, SM., Wang, ZQ. PMHSS iteration method and preconditioners for Stokes control PDE-constrained optimization problems. Numer Algor 87, 365–380 (2021). https://doi.org/10.1007/s11075-020-00970-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00970-1

Keywords

Search

Navigation