In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR m...

We present a new class of iterative schemes for large scale set– valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be either regarded as nonsmooth Newton–type methods for the nonlinear Schur complement or as Uzawa–type iterations with active set preconditioners. Numerical experiments with a control ...

This paper concerns an adaptive finite element method for the Stefan one-phase problem. We derive a parabolic variational inequality using the Duvaut transformation [1]. In each time-step we consider an adaptive algorithm based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive method for the linear problem. We justify the conve...

In this paper, we first provide semi-convergence analysis for a special GPIU(Generalized Parameterized Inexact Uzawa) method with singular preconditioners for solving singular saddle point problems. We next provide a methodology of how to choose nearly quasi-optimal parameters of the special GPIU method. Lastly, numerical experiments are carried out to examine the effectiveness of the special G...

We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by Zhu and Chan in [M. Zhu, and T. F. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA CAM Report [08-34], May 2008], draw connections to similar methods and discuss convergence of several special cases and modifications. In particular, we point out a convergence result fo...

The method of successive approximation is considered for solving the contact elasticity problem which corresponds to the quasivariational Signorini problem. Auxiliary problems with given frictions arising from each external step of this method are solved by the Uzawa method with iterative proximal regularization of the modified Lagrangian functional. Stabilization of the sequence of auxiliary f...

In this note, the generalized parameterized inexact Uzawa method, abbreviated as the GPIU method, for solving the generalized saddle point problems with symmetric positive semi-definite (2,2) block is studied. The convergence of the method is established, which is an extension of the results obtained in a recent paper by Zhou and Zhang (2009) [22]. Crown Copyright 2013 Published by Elsevier Inc...