New Convergence Analysis of GMRES with Weighted Norms, Preconditioning and Deflation, Leading to a New Deflation Space

TL;DR

The paper advances GMRES analysis for large sparse non-Hermitian systems by deriving an explicit convergence bound for weighted, preconditioned, and deflated GMRES (WPD-GMRES) when the Hermitian part of A is positive definite and the weight matches the preconditioner. It introduces a new deflation space built from eigenvectors of the generalized problem $\mathbf{N}\mathbf{z}=\lambda\mathbf{M}\mathbf{z}$, with a bound linking the deflation quality to the spectrum via $|\lambda|>\tau$, and provides two concrete convergence scenarios for HPD preconditioners. The work then demonstrates, through Jordan, Stokes, and convection-diffusion-reaction test problems, that the proposed deflation space can dramatically reduce iteration counts or enable convergence where GMRES fails, while also offering practical guidance for selecting preconditioners and deflation spaces. The results underscore that deflation space design can yield substantial convergence gains and that the new bounds give actionable insight for solver configuration in real-world sparse linear systems.

Abstract

New convergence bounds are presented for weighted, preconditioned, and deflated GMRES for the solution of large, sparse, non-Hermitian linear systems. These bounds are given for the case when the Hermitian part of the coefficient matrix is positive definite, the preconditioner is Hermitian positive definite, and the weight is equal to the preconditioner. The new bounds are a novel contribution in and of themselves. In addition, they are sufficiently explicit to indicate how to choose the preconditioner and the deflation space to accelerate the convergence. One such choice of deflating space is presented, and numerical experiments illustrate the effectiveness of such space.

Figures (9)

• Figure 1: Convergence for the scaled Jordan block example (Section \ref{['sec:Jordan']}). The number $m$ of deflated vectors varies. Increasing $m$ accelerates convergence as predicted by the theory.
• Figure 2: Convergence for the Stokes problem (Section \ref{['sec:Stokes']}). The number $m$ of deflated vectors varies. Increasing $m$ improves convergence as predicted by the theory.
• Figure 3: Example meshes for the convection-diffusion-reaction problem (Section \ref{['sec:CDR']}). Left: 2373 vertices and 4568 triangles. Right: 8643 vertices and 16948 triangles.
• Figure 4: Convection-diffusion-reaction problem (Section \ref{['sec:CDR']}). Left: Solution. Right: Partition into 16 subdomains for $\mathbf{H}_{\text{DD}}$.
• Figure 5: Convection-diffusion-reaction problem (Section \ref{['sec:CDR']}). $|\lambda_{1}|$ to $|\lambda_{3000}|$ in log scale (solution of \ref{['eq:gevpNM']} for $\eta = 1$).

Theorems & Definitions (25)

• Theorem 1
• Lemma 1
• proof
• Definition 1
• Lemma 2
• Remark 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4