Efficient Adaptive Algorithms for an Electromagnetic Coefficient Inverse Problem
This thesis comprises five scientific papers, all of which are focusing on the inverse problem of reconstructing a dielectric permittivity which may vary in space inside a given domain. The data for the reconstruction consist of time-domain observations of the electric field, resulting from a single incident wave, on a part of the boundary of the domain under consideration. The medium is assumed to be isotropic, non-magnetic, and non-conductive. We model the permittivity as a continuous function, and identify distinct objects by means of iso-surfaces at threshold values of the permittivity. Our reconstruction method is centred around the minimization of a Tikhonov functional, well known from the theory of ill-posed problems, where the minimization is performed in a Lagrangian framework inspired by optimal control theory for partial differential equations. Initial approximations for the regularization and minimization are obtained either by a so-called approximately globally convergent method, or by a (simpler but less rigorous) homogeneous background guess. The functions involved in the minimization are approximated with finite elements, or with a domain decomposition method with finite elements and finite differences. The computational meshes are refined adaptively with regard to the accuracy of the reconstructed permittivity, by means of an a posteriori error estimate derived in detail in the fourth paper. The method is tested with success on simulated as well as laboratory measured data.
Parts of work
Larisa Beilina, Nguyen Trung Thành, Michael V. Klibanov and John Bondestam Malmberg. Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity. Inverse problems 30:105007, 2014. ::doi::10.1088/0266-5611/30/10/105007Larisa Beilina, Nguyen Trung Thành, Michael V. Klibanov and John Bondestam Malmberg. Globally convergent and adaptive finite element methods in imaging of buried objects from experimental backscattering radar measurements. Journal of Computational and Applied Mathematics 289:371--391, 2015. ::doi::10.1016/j.cam.2014.11.055John Bondestam Malmberg. A posteriori error estimate in the Lagrangian setting for an inverse problem based on a new formulation of Maxwell's system, volume 120 of Springer Proceedings in Mathematics and Statistics, pages 42--53, Springer, 2015. ::doi::10.1007/978-3-319-12499-5_3John Bondestam Malmberg, and Larisa Beilina. An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations. Manuscript submitted to Applied Mathematics & Information Sciences.John Bondestam Malmberg, and Larisa Beilina. Iterative Regularization and Adaptivity for an Electromagnetic Coefficient Inverse Problem. Manuscript to appear in the Proceedings of the 14th International Conference of Numerical Analysis and Applied Mathematics.
Doctor of Philosophy
Göteborgs universitet. Naturvetenskapliga fakulteten
Department of Mathematical Sciences ; Institutionen för matematiska vetenskaper
Fredagen den 1 september 2017, kl 13.15, Pascal, Matematiska vetenskaper, Chalmers tvärgata3, Göteborg.
Date of defence
Malmberg, John Bondestam
coefficient inverse problem
approximate global convergence
finite element method
a posteriori error analysis