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

TL;DR

This paper analyzes the spectral properties of a symmetric positive definite block preconditioner for double saddle-point systems, demonstrating its effectiveness in solving large-scale PDE-constrained optimization problems.

Contribution

It introduces a spectral analysis framework for a new class of block preconditioners, linking eigenvalues to roots of a cubic polynomial, with practical efficiency validation.

Findings

Eigenvalues characterized by roots of a cubic polynomial

Preconditioner improves convergence in PDE-constrained optimization

Theoretical bounds verified through numerical experiments

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.