A Modified Inexact Uzawa Algorithm for Generalized Saddle Point Problems

Authors: Shu-Xin Miao

Abstract:

In this note, we discuss the convergence behavior of a modified inexact Uzawa algorithm for solving generalized saddle point problems, which is an extension of the result obtained in a recent paper [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171].

Keywords: Saddle point problem, inexact Uzawa algorithm, convergence behavior.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1058799

References:

[1] Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008) 2900-2932.
[2] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005) 1-137.
[3] J.H. Bramble, J.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997) 1072-1092.
[4] Y. Cao, Y. Lin, Y. Wei, Nolinear Uzawa methods for solving nonsym-metric saddle point problems, J. Appl. Math. Comput., 21 (2006) 1-21.
[5] Z.-H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171.
[6] X.-L. Cheng, The inexact Uzawa algorithm for saddle point problem, Appl. Math. Letters, 13 (2000) 1-3.
[7] H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31(1994) 1645-1661.
[8] B. Fischer, R. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math., 38 (1998) 527-543.
[9] G.H. Golub,
[] X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math., 41 (2001) 71-85.
[10] Y. Lin, Y. Wei, A convergence analysis of the nonlinear Uzawa algorithm for saddle point problems, Appl. Math. Letters, 20 (2007) 1094-1098.
[11] Y. Lin, Y. Wei, Fast corrected Uzawa methods for solving symmetric saddle point problems, Calcolo, 43 (2006) 65-82.
[12] S.-X. Miao, K. Wang, On generalized stationary iterative method for solving the saddle point problems, J. Appl. Math. Comput., DOI: 10.1007/s 12190-009-0369-8.
[13] X. Wu, B.P.B. Silva, J.-Y. Yuan, Conjugate gradient method for rank deficient saddle point problems, Numer. Algorithms, 35 (2004) 139-154.
[14] J.H. Yun, S.W. Kim, Generalized stationary iterative method for solving linear systems, Korean J. Comput. Appl. Math., 5 (1998) 341-349.
[15] B. Zheng, Z.-Z. Bai, X. Yang, On Semi-Convergence of Parameterized Uzawa Methods for Singular Saddle Point Problems, Linear Algebra Appl., 431 (2009) 808-817.