Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms

TL;DR

This paper develops an interior point method with preconditioning for efficiently solving PDE-constrained optimization problems that include sparsity terms, demonstrating robustness and computational efficiency across various parameters.

Contribution

It introduces a novel interior point approach combined with preconditioners for PDE-constrained problems with sparsity, improving solution robustness and efficiency.

Findings

Robust performance of the interior point scheme across parameter variations

Preconditioners significantly reduce iteration counts and CPU times

Effective for multiple PDE applications with sparsity constraints

Abstract

PDE-constrained optimization problems with control or state constraints are challenging from an analytical as well as numerical perspective. The combination of these constraints with a sparsity-promoting $\rm L^1$ term within the objective function requires sophisticated optimization methods. We propose the use of an Interior Point scheme applied to a smoothed reformulation of the discretized problem, and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method we introduce fast and efficient preconditioners which enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.