On use of inhomogeneous media for elimination of ill-posedness in the inverse problem
Loading...
Date
2014-04-17
Authors
Feroj, Md Jamil
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This thesis outlines a novel approach to make ill-posed inverse source problem well-posed exploiting inhomogeneous media. More precisely, we use Maxwell fish-eye lens to make scattered field emanating from distinct regions of an object of interest more directive and concentrated onto distinct regions of observation. The object of interest in this thesis is a thin slab placed conformally to the Maxwell fish-eye lens. Focused Green’s function of the background medium results in diagonal dominance of the matrix to be inverted for inverse problem solution. Hence, the problem becomes well-posed. We have studied one-dimensional variation of a very thin dielectric slab of interest having conformal shape to the lens. This method has been tested solving the forward problem using both Mie series and using COMSOL.
Most common techniques for solving inverse problem are full non-linear inversion techniques, such as: distorted Born iterative method (DBIM) and contrast source inversion (CSI). DBIM needs to be regularized at every iteration. In some cases, it converges to a solution, and, in some cases, it does not. Diffraction tomography does not utilize regularization. It is a technique under Born approximation. It eliminates ill-posedness, but it works only for small contrast. Our proposed method works for high contrast and also provides well-posedness.
In this thesis, our objective is to demonstrate inverse source problem and inverse scattering problem are not inherently ill-posed. They are ill-posed because conventional techniques usually use homogeneous or non-focusing background medium. These mediums do not support separation of scattered field. Utilization of background medium for scattered field separation casts the inverse problem in well-posed form.
Description
Keywords
inverse problem, ill-posedness, well-posed formulation, focusing medium