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

Authors

John Bondestam Malmberg, Department of Mathematicals Sciences, Chalmers University of Technology and Gothenburg University, 412 96 Gothenburg, SwedenFollow
Larisa Beilina, Department of Mathematicals Sciences, Chalmers University of Technology and Gothenburg University, 412 96 Gothenburg, SwedenFollow

Author Country (or Countries)

Sweden

Abstract

We consider a coefficient inverse problem to determine the dielectric permittivity in Maxwell’s equations, with data consisting of boundary measurements. The true dielectric permittivity is assumed to belong to an ideal space of very fine finite elements. 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 norm of the error in the reconstructed permittivity is derived. The adaptive algorithm is formulated and 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.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/120101

Recommended Citation

Bondestam Malmberg, John and Beilina, Larisa (2018) "An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations," Applied Mathematics & Information Sciences: Vol. 12: Iss. 1, Article 1.
DOI: http://dx.doi.org/10.18576/amis/120101
Available at: https://digitalcommons.aaru.edu.jo/amis/vol12/iss1/1