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

TL;DR

The paper tackles solving large, sparse two-by-two complex block systems with $\mathbf{A}=\begin{pmatrix}F&-G^*\\ G&F\end{pmatrix}$ where $F$ is HPD and $G$ is PSD but not necessarily Hermitian. It introduces the MPRESB preconditioner $\mathbf{R}$ and shows that applying it reduces the Krylov solver's burden to solving two HPD subsystems with $F+\frac{1}{2}(G+G^*)$, enabling efficient solvability via Cholesky or CG. Spectral analysis demonstrates that eigenvalues of the preconditioned matrix $\mathbf{R}^{-1}\mathbf{Q}$ lie on $\Re(\lambda)=1$ with bounded imaginary parts, and clustering improves when the skew-Hermitian part of $G$ is small; for PDE-constrained optimization problems, the method reduces to solving two SPD systems and a practical two-step algorithm (Alg.1) is provided. Numerical experiments on 2D and 3D FE-discretized PDE problems show MPRESB often outperforms BD and BAS preconditioners, though 3D cases can incur higher overhead due to factorization of complex subsystems. Overall, MPRESB offers a theoretically justified and practically effective preconditioning strategy for non-Hermitian complex linear systems arising in scientific computing.

Abstract

We present a modified version of the PRESB preconditioner for two-by-two block system of linear equations with the coefficient matrix $$\textbf{A}=\left(\begin{array}{cc} F & -G^* G & F \end{array}\right),$$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian positive definite and $G\in\mathbb{C}^{n\times n}$ 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.

Figures (1)

• Figure 1: Eigenvalue distribution of $\mathbf{R}_{MPRESB}^{-1}\mathbf{A}$ in two dimensional case for $h=2^{-4}$ and $\omega=10$ for $\nu=10^{-k}$, $k=2,4,6,8$.

Theorems & Definitions (5)

• theorem 1
• proof
• remark thmcounterremark
• theorem 2
• proof