A modified version of the PRESB preconditioner for a class of non-Hermitian complex systems of linear equations

2024/02/17 by Owe Axelsson, Axelsson, Owe, Dovod Khojasteh Slakuyeh +1
Computer Science · Mathematics · #65F10 #65F50 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2402.11184

openalex publication_date 2024/02/17 · openalex created_date 2024/02/22 · openalex updated_date 2026/07/28

Abstract

We present a modified version of the PRESB preconditioner for two-by-two block system of linear equations with the coefficient matrix A=(F · amp; -G^* G · amp; F), where F∈ℂ^{n× n} is Hermitian positive definite and G∈ℂ^{n× n} is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly utilizing the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner.

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