An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations

TL;DR

This paper develops an adaptive finite element method combined with a Lagrangian approach to improve the quantitative reconstruction of small dielectric inclusions in Maxwell's equations from limited boundary measurements.

Contribution

It introduces a new a posteriori error estimate for the coefficient and demonstrates its effectiveness through numerical experiments involving multiple small inclusions.

Findings

Successful reconstruction of multiple small inclusions with low contrast

Effective handling of superimposed Gaussian functions

Validation of the method's accuracy through numerical tests

Abstract

We consider a coefficient inverse problem for the dielectric permittivity in Maxwell's equations, with data consisting of boundary measurements of one or two backscattered or transmitted waves. The problem is treated using a Lagrangian approach to the minimization of a Tikhonov functional, where an adaptive finite element method forms the basis of the computations. A new a posteriori error estimate for the coefficient is derived. The method is tested successfully in numerical experiments for the reconstruction of two, three, and four small inclusions with low contrast, as well as the reconstruction of a superposition of two Gaussian functions.