Positive semidefinite/positive semidefinite splitting iteration methods for solving nonsingular non-Hermitian positive semidefinite systems
Davod Khojasteh Salkuyeh, Mohsen Masoudi
TL;DR
The paper introduces the Positive semidefinite/Positive semidefinite Splitting (PPS) method for solving nonsingular non-Hermitian positive semidefinite systems by decomposing $\mathcal{A}$ into PSD components and using a HPD shift $\Sigma$ to form an effective preconditioner. It provides rigorous convergence analysis with sufficient conditions and bounds, shows PPS generalizes several known splittings (HSS, PSS, NSS, GHSS), and develops SPPS1 and SPPS2 variants to exploit structured matrices for efficient inner solves. The authors offer practical guidelines for selecting $\Sigma$, and $\mathcal{P}_1$, and $\mathcal{P}_2$, including an optimal $\alpha$ for $\Sigma=\alpha Q$, and demonstrate through numerical experiments that SPPS-based preconditioners yield superior convergence and robustness compared to existing methods, including in image restoration and PDE discretizations. The work thus provides a flexible, scalable framework for efficient iterative solution of a broad class of nonsymmetric positive semidefinite linear systems with clear guidance for implementation.
Abstract
This article introduces an iterative method for solving nonsingular non-Hermitian positive semidefinite systems of linear equations. To construct the iteration process, the coefficient matrix is split into two non-Hermitian positive semidefinite matrices along with an arbitrary Hermitian positive definite shift matrix. Several conditions are established to guarantee the convergence of method and suggestions are provided for selecting the matrices involved in the desired splitting. We explore selection process of the shift matrix and determine the optimal parameter in a specific scenario. The proposed method aims to generalize previous approaches and improve the conditions for convergence theorems. In addition, we examine two special cases of this method and compare the induced preconditioners with some state-of-art preconditioners. Numerical examples are given to demonstrate effectiveness of the presented preconditioners.