Journal ArticleDOI
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
TLDR
This paper considers the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems, and shows that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled.Abstract:
In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.read more
Citations
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Journal ArticleDOI
Numerical solution of saddle point problems
TL;DR: A large selection of solution methods for linear systems in saddle point form are presented, with an emphasis on iterative methods for large and sparse problems.
Journal ArticleDOI
Split Bregman Methods and Frame Based Image Restoration
TL;DR: It is proved the convergence of the split Bregman iterations, where the number of inner iterations is fixed to be one, which gives a set of new frame based image restoration algorithms that cover several topics in image restorations.
Journal ArticleDOI
Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems
TL;DR: A class of preconditioned Hermitian/skew-Hermitian splitting iteration methods is established, showing that the new method converges unconditionally to the unique solution of the linear system.
Journal ArticleDOI
A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration
TL;DR: The convergence of the general algorithm framework is proved under mild assumptions and the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.
Journal ArticleDOI
A Preconditioner for Generalized Saddle Point Problems
Michele Benzi,Gene H. Golub +1 more
TL;DR: A preconditioning strategy based on the symmetric\slash skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditionsed matrix are established.
References
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Journal ArticleDOI
Methods of Conjugate Gradients for Solving Linear Systems
TL;DR: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns and it is shown that this method is a special case of a very general method which also includes Gaussian elimination.
Book
Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.