Discrete empirical interpolation and unfitted mesh FEMs: application in PDE-constrained optimization

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.

Keywords:

• Optimal control
• cut finite element method
• reduced order methods
• geometrical parametrization
• empirical interpolation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This project has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under grant agreement No[1115] (PI: E. Karatzas), the ‘First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment’ grant 3270 and the support of the National Infrastructures for Research and Technology S.A. (GRNET S.A.) in the National HPC facility – ARIS – under project ID pa190902.