Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements

TL;DR

This paper develops and tests advanced finite element and Monte Carlo methods for solving elliptic optimal control problems on uncertain, complex geometries using fictitious domain and cut element techniques, ensuring efficiency and accuracy.

Contribution

It introduces a novel approach combining unfitted FEM, improved Monte Carlo methods, and specialized preconditioners for optimal control problems on random domains with challenging geometries.

Findings

Effective preconditioners improve solver convergence.

Optimal error estimates confirm method accuracy.

Multigrid schemes enhance computational efficiency.

Abstract

This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally "forbidden" combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non--deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi-Monte Carlo method and emphasizing a deterministic target state. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state, and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples.