Discrete Empirical Interpolation and unfitted mesh FEMs: application in PDE-constrained optimization

TL;DR

This paper develops a reduced order modeling approach combining Discrete Empirical Interpolation with CutFEM for efficient PDE-constrained optimization in parametrized domains, ensuring rapid solutions across configurations.

Contribution

It introduces a novel reduced order solver integrating Discrete Empirical Interpolation with CutFEM for PDE-constrained optimization in fixed background geometries.

Findings

High fidelity CutFEM performance demonstrated

Efficient reduced order model achieved with rapid online computations

Effective application to quadratic optimization problems with elliptic constraints

Abstract

In this work, we investigate the performance CutFEM as a high fidelity solver as well as we construct a competent and economical reduced order solver for PDE-constrained optimization problems in parametrized domains that live in a fixed background geometry and mesh. Its effectiveness and reliability will be assessed through its application for the numerical solution of quadratic optimization problems with elliptic equations as constraints, examining an archetypal case. The reduction strategy will be via Proper Orthogonal Decomposition of suitable FE snapshots, using an aggregated state and adjoint test space, while the efficiency of the offline-online decoupling will be ensured by means of Discrete Empirical Interpolation of the optimality system matrix and right-hand side, enabling thus a rapid resolution of the reduced order model for each new spatial configuration.