New modified shift-splitting preconditioners for non-symmetric saddle point problems

Mahin Ardeshiry,Hossein Noormohammadi Pour,Hossein Sadeghi Goughery

doi:10.1007/s40065-019-0256-6

Abstract

Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shift-splitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

Highlights

Consider the following non-symmetric saddle point linear system AU = AB −BT 0 x y = f −g = b, (1)where A ∈ Rn×n is positive definite; B ∈ Rn×m(m ≤ n) is a rectangular matrix of rank r ≤ m; f ∈ Rn and g ∈ Rm are the given vectors.In general, matrices A and B in A are large and sparse
We present new modified shift-splitting (NMSS) preconditioners for this type of the saddle point problems (1)
If λ is an eigenvalue of iteration matrix Γ and u = [xT, yT]T is the eigenvector of Γ corresponding to λ, NMSS iteration method converges to the unique solution of problem (1) if and only if parameters α and β satisfy

Summary

Introduction

Where A ∈ Rn×n is positive definite (symmetric or non-symmetric); B ∈ Rn×m(m ≤ n) is a rectangular matrix of rank r ≤ m; f ∈ Rn and g ∈ Rm are the given vectors. Iterative methods are used for solving saddle point problems (1), when matrix blocks A and B are large and sparse Some of these methods, such as Uzawa [7], inexact Uzawa [16] and the Hermitian and skew-Hermitian splitting method [2,18,21] have been presented. Zhou et al [26] and Huang et al [17], respectively, proposed modified shift-splitting (MSS) and generalized modified shift-splitting (GMSS) preconditioners, for solving non-Hermitian saddle point problems. They used symmetric and skew-symmetric splitting of the (1, 1)-block A to make these preconditioners. Practical numerical examples are presented to show the effectiveness of the NMSS preconditioners

New modified shift-splitting method

Convergence of NMSS iteration method

T BTU βI

Preconditioning properties

Numerical results

Conclusion