| Inverse problems of parameter identification and source identification in partial differential equations are highly ill-posed problems and for their satisfactory theoretical and numerical treatment some sort of regularization is necessary. In this thesis, we pose this inverse problem as an optimization problem and perform the regularization in Tikhonov sense. The most crucial aspect of the study of the regularized optimization problem is a proper selection of the regularization parameter. Although the theory for one of the most efficient methods for choosing an optimal regularization parameter, the so-called Morozov discrepancy principle, is well-developed for linear inverse problems, its use for nonlinear inverse problems is rather heuristic. In this thesis, we investigate the inverse problem of parameter identification using an equation error approach in which the coefficient appear linearly. Using the results known for linear inverse problems, we develop a rigorous Morozov discrepancy principle for nonlinear inverse problems. We present a detailed computational experimentation to test the feasibility of the developed approach. We also study the inverse problem of source identification in fourth-order boundary value problem. |
