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Field of values analysis that includes zero for preconditioned nonsymmetric saddle-point systems

Hao Chen, Chen Greif

TL;DR

This work extends field-of-values (FOV) analysis to preconditioned nonsymmetric saddle-point systems where the origin may lie in the FOV by using the spectral set $\Omega_{CG} = W_H(A) \cap \{ z : |z| \ge \|A^{-1}\|_H^{-1} \}$, following the idea that a region with a circular hole is a spectral set. It establishes concrete, dimension-independent GMRES convergence conditions in terms of $a,b,c$ with $\|A\|_H \le a$, $\|A^{-1}\|_H \le b$, and $\|\frac{HA - A^T H}{2}\|_{H,H^{-1}} \le c$ with $bc<1$, and applies these results to block-triangular and block-diagonal preconditioners under the assumption that the skew-symmetric part is small, quantified by a parameter $\alpha$. The theory is validated numerically on Navier–Stokes and Stokes–Darcy problems, showing near-constant iteration counts across mesh refinements for both diagonal and upper-triangular preconditioners under left preconditioning, with explicit $b$ and $c$ values consistent with $bc<1$. The study highlights both the potential of FOV-based guarantees in practical PDE-derived saddle-point systems and the limitations imposed by requiring a small skew-symmetric component, suggesting avenues for refining the geometric FOV analysis and extending to broader preconditioners.

Abstract

We present a field-of-values (FOV) analysis for preconditioned nonsymmetric saddle-point linear systems, where zero is included in the field of values of the matrix. We rely on recent results of Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087-1001, 2019], showing that a convex region with a circular hole is a spectral set. Sufficient conditions are derived for convergence independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems, and show their applicability to families of block preconditioners that have not been previously covered by existing FOV analysis. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity coefficient. Some numerical results illustrate our findings.

Field of values analysis that includes zero for preconditioned nonsymmetric saddle-point systems

TL;DR

This work extends field-of-values (FOV) analysis to preconditioned nonsymmetric saddle-point systems where the origin may lie in the FOV by using the spectral set , following the idea that a region with a circular hole is a spectral set. It establishes concrete, dimension-independent GMRES convergence conditions in terms of with , , and with , and applies these results to block-triangular and block-diagonal preconditioners under the assumption that the skew-symmetric part is small, quantified by a parameter . The theory is validated numerically on Navier–Stokes and Stokes–Darcy problems, showing near-constant iteration counts across mesh refinements for both diagonal and upper-triangular preconditioners under left preconditioning, with explicit and values consistent with . The study highlights both the potential of FOV-based guarantees in practical PDE-derived saddle-point systems and the limitations imposed by requiring a small skew-symmetric component, suggesting avenues for refining the geometric FOV analysis and extending to broader preconditioners.

Abstract

We present a field-of-values (FOV) analysis for preconditioned nonsymmetric saddle-point linear systems, where zero is included in the field of values of the matrix. We rely on recent results of Crouzeix and Greenbaum [Spectral sets: numerical range and beyond. SIAM Journal on Matrix Analysis and Applications, 40(3):1087-1001, 2019], showing that a convex region with a circular hole is a spectral set. Sufficient conditions are derived for convergence independent of the matrix dimensions. We apply our results to preconditioned nonsymmetric saddle-point systems, and show their applicability to families of block preconditioners that have not been previously covered by existing FOV analysis. A limitation of our theory is that the preconditioned matrix is required to have a small skew-symmetric part in norm. Consequently, our analysis may not be applicable, for example, to fluid flow problems characterized by a small viscosity coefficient. Some numerical results illustrate our findings.
Paper Structure (13 sections, 18 theorems, 77 equations, 4 figures, 2 tables)

This paper contains 13 sections, 18 theorems, 77 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. FOV Analysis that Includes Zero
  3. Preliminaries
  4. Sufficient Conditions
  5. Scope and Limitations of the Analysis
  6. Preconditioned Saddle-Point Matrices with Zero in the Field of Values
  7. Block-Triangular Preconditioners
  8. A Block-Diagonal Preconditioner
  9. Inexact Preconditioning
  10. Numerical Experiments
  11. Navier-Stokes Equations
  12. Stokes-Darcy Equations
  13. Concluding Remarks

Key Result

Theorem 1.3

crouzeix2017numerical Let $A$ be a matrix of the same dimensions as $H$. Then, $W_H(A)$ is a $(1+\sqrt{2})$-spectral set for $A$.

Figures (4)

  • Figure 1: The shaded region is $\Omega_{D}$ when conditions \ref{['cond1']}--\ref{['cond4']} of Lemma \ref{['lem:fov_main']} hold
  • Figure 2: The shaded region is $\Omega_{D}$ when $bc \ge 1$ (i.e., when condition \ref{['cond4']} of Lemma \ref{['lem:fov_main']} is violated)
  • Figure 3: The shaded region is $\Omega_{\text{FOV}}$ with $\alpha = 0.5$ and $\beta = 1$
  • Figure 4: The shaded region is $\Omega_{CG}$ for $A$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2: loghin2004analysis
  • Theorem 2.3: driscoll1998potential
  • Lemma 2.4
  • proof
  • ...and 28 more