An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations
For researchers in inverse problems and computational electromagnetics, this work provides an adaptive method for reconstructing small inclusions from limited observations, but the results are incremental and lack comparison to existing methods.
The paper tackles a coefficient inverse problem for dielectric permittivity in Maxwell's equations using limited boundary measurements. The proposed adaptive finite element method successfully reconstructs small inclusions with low contrast in numerical experiments.
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.