A new preconditioner for elliptic PDE-constrained optimization problems

TL;DR

This paper introduces a novel preconditioner designed to improve the convergence speed of GMRES when solving linear systems from PDE-constrained optimization problems, demonstrating superior efficiency through numerical comparisons.

Contribution

A new preconditioner tailored for PDE-constrained optimization problems that enhances GMRES convergence, with analysis of eigenvalues and eigenvectors.

Findings

Preconditioner accelerates GMRES convergence.

Eigenvalue distribution analysis supports effectiveness.

Numerical results outperform existing preconditioners.

Abstract

We propose a preconditioner to accelerate the convergence of the GMRES iterative method for solving the system of linear equations obtained from discretize-then-optimize approach applied to optimal control problems constrained by a partial differential equation. Eigenvalue distribution of the preconditioned matrix as well as its eigenvectors are discussed. Numerical results of the proposed preconditioner are compared with several existing preconditioners to show its efficiency.