Variational optimization approach for reconstruction of dielectric permittivity and conductivity functions using partial boundary measurements

TL;DR

This paper introduces a variational optimization method for reconstructing dielectric permittivity and conductivity in Maxwell's equations using limited boundary data, supported by theoretical analysis and numerical validation.

Contribution

It develops a novel variational framework with finite element algorithms for simultaneous reconstruction of dielectric properties from boundary measurements.

Findings

The method successfully reconstructs dielectric permittivity and conductivity in simulations.

Theoretical stability and differentiability properties are established.

Numerical results confirm the approach's effectiveness in 2D and 3D cases.

Abstract

We present a variational optimization approach for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and conductivity functions in time-dependent Maxwell's system using limited boundary observations of the electric field. The variational optimization approach is based on constructing a weak form of a Lagrangian which allows to use finite element based reconstruction algorithms. The optimality conditions for the Lagrangian and stability estimate for the adjoint problem are derived, as well as Frech\'et differentiability of it and of the regularized Tikhonov functional are also presented. Two- and three-dimensional numerical studies confirm our theoretical investigations.