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Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems

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Abstract

In this work, we consider numerical methods for solving a class of block three-by-three saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented.

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References

  1. Aasen, J. O.: On the reduction of a symmetric matrix to tridiagonal form. BIT Numer. Math. 11, 233–242 (1971)

    MathSciNet  MATH  Google Scholar 

  2. Axelsson, O.: Unified analysis of preconditioning methods for saddle point matrices. Numer. Linear Algebra Appl. 22, 233–253 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Arridge, S. R.: Optical tomography in medical imaging. Inverse Probl. 15, 41–93 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Stanford University Press, Stanford (1958)

    MATH  Google Scholar 

  5. Arioli, M., Manzini, G.: A null space algorithm for mixed finite-element approximations of Darcy’s equation. Comm. Numer. Meth. Engrg. 18, 645–657 (2002)

    MATH  Google Scholar 

  6. Benzi, M.: Solution of equality-constrained quadratic programming problems by a projection iterative method. Rend. Mat. Appl. 13, 275–296 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Bertsekas, D. P.: Nonlinear programming, 2nd edn. Athena Scientific, Massachusetts (1999)

  8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991)

    MATH  Google Scholar 

  9. Bai, Z. -Z., Golub, G. H., Lu, L. -Z., Yin, J. -F.: Block triangular and skew-Hermitian splitting method for positive-definite linear systems. SIAM J. Sci Comput. 26, 844–863 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Bai, Z. -Z., Golub, G. H., Ng, M. K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Bai, Z. -Z., Golub, G. H., Pan, J. -Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Bunch, J. R., Parlett, B. N.: Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639–655 (1971)

    MathSciNet  Google Scholar 

  13. Bramble, J. H., Pasciak, J. E., Vassilev, A. T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072–1092 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Bai, Z. Z., Wang, Z. Q.: On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl. 428, 2900–2932 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Bank, R. E., Welfert, B. D., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numer. Math. 56, 645–666 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Chen, X.: On preconditioned Uzawa methods and SOR methods for saddle-point problems. J. Comput. Appl. Math. 100, 207–224 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Christiansen, S. H.: Discrete Fredholm properties and convergence estimates for the electric field integral equation. Math Comput. 73, 143–167 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Chen, Z. -M., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer Anal. 37, 1542–1570 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Chen, F., Jiang, Y. -L.: A generalization of the inexact parameterized Uzawa methods for saddle point problems. Appl. Math. Comput. 206, 765–771 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Dyn, N., Ferguson, W. E.: The numerical solution of equality constrained quadratic programming problems. Math. Comp. 41, 165–170 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Duff, I. S., Gould, N. I. M., Reid, J. K., Scott, J. A., Turner, K.: The factorization of sparse symmetric indefinite matrices. IMA J. Numer. Anal. 11, 181–204 (1991)

    MathSciNet  MATH  Google Scholar 

  22. Day, D. D., Heroux, M. A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23, 480–498 (2001)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Elman, H. C., Ramage, A., Silvester, D. J.: Algorithm 866: IFISS, a MATLAB toolbox for modelling incompressible flow. ACM Trans. Math. Software 33, 1–18 (2007)

    MATH  Google Scholar 

  25. Forsgren, A.: Inertia-controlling factorizations for optimization algorithms. Appl. Numer. Math. 43, 91–107 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Gould, N. I. M., Hribar, M. E., Nocedal, J.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23, 1376–1395 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Greif, C., Moulding, E., Orban, D.: Bounds on eigenvalues of matrices arising from interior-point methods. SIAM J. Optim. 24, 49–83 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Gould, N. I. M., Orban, D., Toint, P. L.: CUTEr and SifDec, a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29, 373–394 (2003)

    MATH  Google Scholar 

  29. Golub, G. H., Wathen, A. J.: An iteration for indefinite systems and its application to the Navier-Stokes equations. SIAM J. Sci. Comput. 19, 530–539 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Golub, G. H., Wu, X., Yuan, J. -Y.: SOR-like methods for augmented systems. BIT Numer. Math. 41, 71–85 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Householder, A.S.: The theory of matrices in numerical analysis, Blaisdell (1964)

  32. Huang, N., Ma, C. -F.: Spectral analysis of the preconditioned system for the 3 × 3 block saddle point problem, Numer Algor. https://doi.org/10.1007/s11075-018-0555-6(2018)

  33. Huai, K. -Z., Ni, M. -F., Yu, Z. -K., Zheng, X., Ma, F.: A generalized inexact Uzawa method for stable principal component pursuit problem with nonnegative constraints. Numer. Algor. 77, 653–674 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49, 409–436 (1952)

    MathSciNet  MATH  Google Scholar 

  35. Han, D. -R., Yuan, X. -M.: Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J. Numer. Anal. 51, 3446–3457 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Hu, Q. -Y., Zou, J.: An iterative method with variable relaxation parameters for saddle-point problems. SIAM J. Matrix Anal. Appl. 23, 317–338 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Hu, Q. -Y., Zou, J.: Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems. Numer. Math. 93, 333–359 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Hu, Q. -Y., Zou, J.: Nonlinear inexact Uzawa algorithms for linear and nonlinear saddle-point problems. SIAM J. Optim. 16, 798–825 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Jones, M. T., Patrick, M. L.: Bunch-Kaufman factorization for real symmetric indefinite banded matrices. SIAM J. Matrix Anal. Appl. 14, 553–559 (1993)

    MathSciNet  MATH  Google Scholar 

  40. Liang, Z. -Z., Zhang, G. -F.: On block-diagonally preconditioned accelerated parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 221, 89–101 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Morini, B., Simoncini, V., Tani, M.: Spectral estimates for unreduced symmetric KKT systems arising from interior point methods. Numer. Linear Algebra Appl. 23, 776–800 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Morini, B., Simoncini, V., Tani, M.: A comparison of reduced and unreduced KKT systems arising from interior point methods. Comput. Optim. Appl. 68, 1–27 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Nochetto, R. H., Pyo, J. H.: Optimal relaxation parameter for the Uzawa method. Numer. Math. 98, 695–702 (2004)

    MathSciNet  MATH  Google Scholar 

  44. Parlett, B. N., Reid, J. K.: On the solution of a system of linear equations whose matrix is symmetric but not definite. BIT Numer. Math. 10, 386–397 (1970)

    Google Scholar 

  45. Paige, C. C., Saunders, M. A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)

    MathSciNet  MATH  Google Scholar 

  46. Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7, 585–616 (2000)

    MathSciNet  MATH  Google Scholar 

  47. Wang, N. -N., Li, J. -C., Li, G., Kong, X.: Variants of the Uzawa method for three-order block saddle point problem. Appl. Math. Comput. 305, 188–202 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Sarin, V., Sameh, A.: An efficient iterative method for the generalized Stokes problem. SIAM J. Sci. Comput. 19, 206–226 (1998)

    MathSciNet  MATH  Google Scholar 

  49. Varga, R. S.: Matrix Iterative Analysis. Prentice-Hall, Englewood (1962)

    MATH  Google Scholar 

  50. Van der Vorst, H.: Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  51. Young, D. M.: Iterative Solution for Large Linear Systems. Academic press, New York (1971)

    Google Scholar 

  52. Yang, A. -L., Wu, Y. -J.: The Uzawa-HSS method for saddle-point problems. Appl. Math. Lett. 38, 38–42 (2014)

    MathSciNet  MATH  Google Scholar 

  53. Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71, 479–505 (2002)

    MathSciNet  MATH  Google Scholar 

  54. Zhang, G. -F., Wang, S. -S.: A generalization of parameterized inexact Uzawa method for singular saddle point problems. Appl. Math. Comput. 219, 4225–4231 (2013)

    MathSciNet  MATH  Google Scholar 

  55. Zhang, G. -F., Yang, J. -L., Wang, S. -S.: On generalized parameterized inexact Uzawa method for a block two-by-two linear system. J. Comput. Appl. Math. 255, 193–207 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank the referees for a number of very helpful and insightful comments and suggestions.

Funding

The project was supported by National Postdoctoral Program for Innovative Talents (Grant No.BX201600182), China Postdoctoral Science Foundation (Grant No.2016M600141).

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  1. Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, China

    Na Huang

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  1. Na Huang
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Correspondence to Na Huang.

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Huang, N. Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems. Numer Algor 85, 1233–1254 (2020). https://doi.org/10.1007/s11075-019-00863-y

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