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Spectral Analysis of Block Diagonally Preconditioned Multiple Saddle-Point Matrices with Inexact Schur Complements

Marco Pilotto, Luca Bergamaschi, Angeles Martinez

TL;DR

The paper develops a polynomial-based framework to bound the eigenvalues of block-diagonal Schur-complement preconditioned multiple saddle-point systems with inexact Schur complements, valid for an arbitrary number of blocks $N$. By linking the spectrum to zeros of a monic sequence of Chebyshev-like polynomials $U_k(\\lambda,\\boldsymbol{\\gamma}_E,\\boldsymbol{\\gamma}_R)$ and introducing the parameter rays $\\gamma_E^{(i)},\\gamma_R^{(i)}$, the authors derive interlacing properties, extremal-zero bounds, and explicit eigenvalue intervals. They further show how zeros move with the parameters, enabling evaluation of bounds at endpoint values, and provide concrete bounds for the double saddle-point case $N=2$. Numerical experiments on random and realistic Biot-type problems validate the theoretical bounds and demonstrate the effectiveness of inexact block-diagonal preconditioning in MINRES. The results offer rigorous, computable spectral estimates that inform preconditioner design for multiphysics systems with multiple saddle points.

Abstract

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where the Schur complements are approximated, generalizing the findings in [Bergamaschi et al., Linear Algebra and its Applications, 2026]. Numerical experiments are carried out to validate the proposed estimates.

Spectral Analysis of Block Diagonally Preconditioned Multiple Saddle-Point Matrices with Inexact Schur Complements

TL;DR

The paper develops a polynomial-based framework to bound the eigenvalues of block-diagonal Schur-complement preconditioned multiple saddle-point systems with inexact Schur complements, valid for an arbitrary number of blocks . By linking the spectrum to zeros of a monic sequence of Chebyshev-like polynomials and introducing the parameter rays , the authors derive interlacing properties, extremal-zero bounds, and explicit eigenvalue intervals. They further show how zeros move with the parameters, enabling evaluation of bounds at endpoint values, and provide concrete bounds for the double saddle-point case . Numerical experiments on random and realistic Biot-type problems validate the theoretical bounds and demonstrate the effectiveness of inexact block-diagonal preconditioning in MINRES. The results offer rigorous, computable spectral estimates that inform preconditioner design for multiphysics systems with multiple saddle points.

Abstract

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where the Schur complements are approximated, generalizing the findings in [Bergamaschi et al., Linear Algebra and its Applications, 2026]. Numerical experiments are carried out to validate the proposed estimates.
Paper Structure (9 sections, 11 theorems, 71 equations, 3 figures, 4 tables)

This paper contains 9 sections, 11 theorems, 71 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Eigenvalues of Preconditioned Saddle-Point Systems with inexact Schur complements
  3. Bounds for the extremal zeros of the sequence \ref{['eq2']}
  4. How zeros of $\{U_k\}$ move depending on $\gamma_E^{(j)}, \gamma_R^{(j)}$
  5. Bounds for the eigenvalues of the preconditioned matrix
  6. Numerical experiments: Randomly-generated matrices.
  7. Numerical experiments: A realistic test case
  8. Handling the case $n_2 > n_1$
  9. Conclusions

Key Result

Proposition 1

Since the polynomials of the sequence eq2 are monic

Figures (3)

  • Figure 1: Double saddle-point linear system. Extremal eigenvalues of the preconditioned matrix (blue dots) and bounds (red line) after $10$ runs with each combination of the parameters from Table \ref{['tab:1']}.
  • Figure 2: Triple saddle-point linear system. Extremal eigenvalues of the preconditioned matrix (blue dots) and bounds (red line) after $10$ runs with each combination of the parameters from Table \ref{['tab:1']}.
  • Figure 3: Quadruple saddle-point linear system. Extremal eigenvalues of the preconditioned matrix (blue dots) and bounds (red line) after $10$ runs with each combination of the parameters from Table \ref{['tab:1']}.

Theorems & Definitions (23)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • ...and 13 more