Inexact and preconditioned Uzawa algorithms for saddle point problems. (English) Zbl 0815.65041

Elman, Howard C.; Golub, Gene H.

SIAM J. Numer. Anal. 31, No. 6, 1645-1661 (1994).

This paper develops and analyzes variants of the classical Uzawa algorithm for solving symmetric indefinite linear systems, which arise frequently from minimization of quadratic forms subject to linear constraints. Generally, the Uzawa algorithm consists of two parts, the so-called “inner” and “outer” iterations. The “inner” iteration requires the solution of a linear system with symmetric definite coefficient matrix.
The primary concern in this paper is to show that even though this system is not solved exactly, the resulting “inexact” Uzawa algorithm can also be made to be convergent, only with relatively modest requirements on the accuracy of the approximative solution.
The results presented here answer a long-standing question concerning the possibility of using inexact solutions for the Uzawa algorithm.

Reviewer: Liu Xinguo (Qingdao)

MSC:

65F10	Iterative numerical methods for linear systems
65K05	Numerical mathematical programming methods
90C20	Quadratic programming

Keywords:

saddle point problems; outer iterations; inner iteration; inexact Uzawa algorithm; preconditioner; symmetric indefinite linear systems; minimization of quadratic forms; inexact solutions