Sparsity Constraints and Regularization for Nonlinear Inverse Problems

Abstract

Sparsity regularization method has been analyzed for linear and nonlinear inverse problems over the last years. The method is known to be simple for use and has many advantages for problems with sparse solutions. It has been well-developed for linear inverse problems. However, there have been few results proposed for nonlinear inverse problems. Recently, some numerical algorithms for the method have been introduced. Most of them are known to have a linear convergence rate and to be slow in practice, especially for nonlinear inverse problems. The subject of the thesis is to investigate sparsity regularization for nonlinear inverse problems. We aim at the following fields: First, the method is explored for the diffusion coefficient identification problem and electrical impedance tomography. In these problems, the energy functional approach (incorporating with sparsity regularization) is applied instead of the least squares approach. We will analyze advantages of the new approach as well as the well-posedness and some convergence rates of the method in each problem. Second, we propose numerical algorithms for minimization problems in sparsity regularization of nonlinear inverse problems. They consist of a gradient-type method, two accelerated versions, and semi-smooth Newton and quasi-Newton methods. We concentrate on the convergence of the methods. However, for some algorithms, the convergence rate as well as the decreasing rate of the objective functionals are also concerned. The algorithms are then carried out to two parameter identification problems above and the efficiency of the algorithms are examined and illustrated by some specfific examples.