Discrete regularization for parameter identification problems
Inverse problems in the context of partial differential equations play a crucial role in numerous applications ranging from medical imaging via nondestructive testing to geophysical prospecting. Specifically, parameter identification problems, in principle, are to identify a best estimate of the values of one or more parameters in a regression. In recent decades, regularization theory has received a lot of attention due to its vital importance in recovering the values of true solution for inverse problems as well as its applications in numerous different areas of study. However, of particular interest in this thesis are inverse topology problems (e.g., segmentation in medical imaging problems) in which the sought-for distributed parameter can be assumed to take values only from a known finite set. Since the parameter belongs to an infinite-dimensional (or, after discretization, a very large-dimensional) space, combinatorial methods for the numerical solution of such problems are infeasible. In this thesis, a particular regularization strategy is proposed, which is able to recover parameters for a class of inverse topology problems described above using properties of known finite set, namely "multi-bang regularization". Specifically, a function named "multi-bang" is considered as a regularization term in Tikhonov functional which is used to recover parameters for discrete-valued inverse problems. The properties of this term ensure well-posedness of the proposed regularization method. In the main part of this thesis, a convergence analysis of multi-bang regularization is studied. Additionally, clear illustrations of the approach are provided for either linear or nonlinear inverse problems in the form of numerical experiment employing a semi-smooth Newton method and a Moreau-Yosida approximation which is popular for coping with non single-valued functions. "Nice" obtained reconstructions strongly demonstrate the effectiveness of multi-bang regularization in recovering solutions for discrete-valued inverse problems. The regularity of reconstructions is promoted by combining multi-bang penalty term with total variation which leads to an employment of Chambolle-Pock algorithm for solving numerical approximation effectively.