A low-rank matrix equation method for solving PDE-constrained optimization problems

TL;DR

This paper introduces a novel low-rank matrix equation approach to efficiently solve large-scale PDE-constrained optimization problems by reformulating the KKT system into a Sylvester-like matrix equation and applying an iterative Krylov method.

Contribution

The paper presents a new framework that reformulates PDE-constrained optimization problems into a Sylvester-like matrix equation and solves it efficiently with a low-rank approximation using Krylov methods.

Findings

The method effectively computes low-rank solutions for large PDE-constrained optimization problems.

Numerical experiments demonstrate competitive performance compared to existing low-rank approaches.

The approach reduces computational complexity and memory requirements for large-scale problems.

Abstract

PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment or blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible. We present a new framework to efficiently compute a low-rank approximation to the solution by reformulating the KKT system into a Sylvester-like matrix equation. This matrix equation is subsequently projected onto a small subspace via an iterative rational Krylov method and we obtain a reduced problem by imposing a Galerkin condition on its residual. In our work we discuss implementation details and dependence on the various problem parameters. Numerical experiments illustrate the performance of the new strategy also when compared to other low-rank approaches.