Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral analysis of block preconditioners for double saddle-point linear systems with application to PDE-constrained optimization

Luca Bergamaschi, Angeles Martinez, John Pearson, Andreas Potschka

TL;DR

The paper analyzes a symmetric SPD block preconditioner for double saddle-point systems arising in PDE-constrained optimization, showing that the spectrum of the preconditioned matrix is controlled by roots of real-coefficient polynomials $p(\\lambda; \\gamma_A, \\gamma_R)$ and its extension $\\pi_E(\\lambda; \\gamma_A, \\gamma_R, \\gamma_K, \\gamma_E)$. It develops tight eigenvalue bounds, including refinements when the off-diagonal block $C$ is invertible, and handles cases with $E=0$ and $E eq 0$ within a unified polynomial framework. The results are supported by numerical experiments on full and boundary PDE observation problems, where Chebyshev semi-iterations and algebraic multigrid approximations yield mesh- and parameter-robust bounds that predict MINRES convergence. Overall, the work provides a rigorous, spectrum-based diagnostic for preconditioner effectiveness in large-scale PDE-constrained optimization settings.

Abstract

In this paper, we describe and analyze the spectral properties of a symmetric positive definite inexact block preconditioner for a class of symmetric, double saddle-point linear systems. We develop a spectral analysis of the preconditioned matrix, showing that its eigenvalues can be described in terms of the roots of a cubic polynomial with real coefficients. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds, in solving large-scale PDE-constrained optimization problems.

Spectral analysis of block preconditioners for double saddle-point linear systems with application to PDE-constrained optimization

TL;DR

The paper analyzes a symmetric SPD block preconditioner for double saddle-point systems arising in PDE-constrained optimization, showing that the spectrum of the preconditioned matrix is controlled by roots of real-coefficient polynomials and its extension . It develops tight eigenvalue bounds, including refinements when the off-diagonal block is invertible, and handles cases with and within a unified polynomial framework. The results are supported by numerical experiments on full and boundary PDE observation problems, where Chebyshev semi-iterations and algebraic multigrid approximations yield mesh- and parameter-robust bounds that predict MINRES convergence. Overall, the work provides a rigorous, spectrum-based diagnostic for preconditioner effectiveness in large-scale PDE-constrained optimization settings.

Abstract

In this paper, we describe and analyze the spectral properties of a symmetric positive definite inexact block preconditioner for a class of symmetric, double saddle-point linear systems. We develop a spectral analysis of the preconditioned matrix, showing that its eigenvalues can be described in terms of the roots of a cubic polynomial with real coefficients. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds, in solving large-scale PDE-constrained optimization problems.
Paper Structure (9 sections, 13 theorems, 124 equations, 5 figures, 10 tables)

This paper contains 9 sections, 13 theorems, 124 equations, 5 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Eigenvalue analysis of an inexact SPD preconditioner
  3. Spectral analysis in a simplified case
  4. Eigenvalue bounds with $E \ne 0$
  5. Refined upper bound when $C$ is invertible.
  6. Numerical Experiments
  7. Full Observation Problem
  8. Boundary Observation Problem
  9. Concluding Remarks

Key Result

Lemma 2.1

Let $Z$ be a symmetric matrix valued function defined in $F \subset \mathbb{R}$, and where $\sigma(Z(\zeta))$ denotes the spectrum of $Z(\zeta)$. Then, for arbitrary $s \neq 0$, there exists a vector $v \neq 0$ such that

Figures (5)

  • Figure 1: Summary of the signs of the relevant quantities for the proof of Lemma \ref{['Lem:optpi']}.
  • Figure 2: Qualitative plots of $\pi(\lambda)$ and $p(\lambda)$ with $\gamma_A=1.639$, $\gamma_R=0.734$, and $\gamma_K=0.251$.
  • Figure 3: Extremal eigenvalues of the preconditioned matrix (blue dots) and bounds obtained from \ref{['bounds_0']} (red line) after 25 runs with each combination of the parameters from \ref{['TabPar']}.
  • Figure 4: Polynomials $\pi(\lambda)$, $-\gamma_E\, p(\lambda)$, and $\pi_E(\lambda)$ with the same values of $\gamma_A, \gamma_R$, and $\gamma_K$ as in Figure 2, and $\gamma_E = 0.512$.
  • Figure 5: Comparisons between the bounds based on $\lambda_-$ (red line), and the refined upper bounds from \ref{['Cor:m=p']} (yellow line) for the negative eigenvalues. Case with $E \equiv 0$ and a square invertible matrix $C$ (compare with \ref{['eigvsbounds']}, top-left plot).

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • Remark 2.1: Notation
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.1
  • ...and 17 more