Approximate inverse for linear and non-linear inverse problems

Abstract

Many applications in science, engineering, medical imaging and industry arise, where specific results need to be inferred from a given set of observations. Such a task is called an inverse problem, which can be expressed by an operator equation of the form Af = g, where f is the unknown quantity under investigation, g is the measurement data, while A is a bounded operator, illustrating the relation between f and g. An added difficulty when trying to solve such problems is the fact that in most cases, the operator A is ill-posed in the Hadamard sense, meaning that the solution of such problems either does not exist, or is not unique, or else does not depend continuously on the data. As a result, various regularization techniques such as the method of approximate inverse were developed to overcome such issues. The aim of this dissertation is to analyze how the approximate inverse method can be used to reconstruct the unknown quantity f in the case of both linear and non-linear inverse problems. The focus of this dissertation will then shift to an in-depth analysis of a practical realization of the inverse conductivity problem, namely Electrical Impedance Tomography (EIT), in which the unknown complex admittivity has to be reconstructed. It will also be shown how the approximate inverse method can be applied to this same problem in order to reconstruct the unknown admittivity.