Optimal Control of Parameterized Maxwell's System: Reduced Basis, Convergence Analysis, and A Posteriori Error Estimates

TL;DR

This paper develops a reduced basis method for parameterized Maxwell's control problems, providing convergence analysis and a posteriori error estimates to efficiently solve high-dimensional problems with parameter variations.

Contribution

It introduces a reduced basis approach for Maxwell's optimal control problems with convergence guarantees and error estimation, advancing computational efficiency and reliability.

Findings

Uniform convergence of reduced solutions under dense parameter sampling

Development of an a posteriori error estimator based on residuals

Effective reduction of computational complexity for parameterized Maxwell's problems

Abstract

We consider control constrained optimal control problems governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept to discretize the control. We create a reduced basis method for the optimal control problem and establish uniform convergence of the reduced order solutions to that of the original high dimensional problem provided that the snapshot parameter sample is dense in the parameter set, with an appropriate parameter separability rule. Finally, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functions in terms of the state and adjoint residuals.