1 Department of Informatics and Mathematical Modeling, Technical University of Denmark
The aim of this thesis is to formalize the process of constructing a numerical regularization method for nonlinear problems. Moreover, to give an intuitive presentation of a discrete nonlinear ill-posed problem. We are able to distinguish three phases in the process of constructing a numerical regularization method: the first phase is where we choose a regularization formulation, based upon the prior information and its mathematical formulation, the second phase is where we choose a numerical method for solving the regularized problem, and the third phase is where we combine the regularization method and the numerical solver into an implementation. By looking at considerations done in the last phase we are able to classify numerical regularization methods into global and local regularization methods. The regularization method we study is the Tikhonov regularization method. We show that the IRGN method and the Levenberg-Marquardt method can be viewed as being equal in the first and second phase, but different considerations in phase three lead to two different methods. Therefore, it is interesting to study and compare the regularization properties of these two methods. We carried out the study with the help of the (G)SVD-analysis of the subproblems solved in each iteration of the two methods. We find that in general, the IRGN method and the LM method are not equivalent, and that the regularization properties of the IRGN method is more consistent with the Tikhonov regularization method, which is the regularization used in both numerical regularization method. Moreover, by separating the phases it is possible to suggest a new approach for using the L-curve for updating the regularization parameter in the IRGN method. We find that the new updating strategy works well for solving the nonlinear Hammerstein integral equation, as well as for solving the two geophysical inverse problems considered in this thesis. We compare the IRGN method, the Levenberg-Marquardt method, the trust-region method and the inexact Gauss-Newton method for solving the nonlinear Hammerstein integral equation, and for solving two geophysical inverse problems: a seismic tomography problem, and a geoelectrical sounding problem. We found that all four methods gave reasonable solutions for the two geophysical problem. However, the inexact Gauss-Newton method converged faster than the others for the seismic tomography problem, and the inexact Gauss-Newton method and the IRGN method work better than the other two for the geoelectrical sounding problem. However, for the nonlinear Hammerstein integral equation the inexact Gauss-Newton method diverges, and the IRGN method converges fastest. The work presented here also considers linear ill-posed problems. We find that a for a special class of discrete linear ill-posed problems, we can calculate approximations to the singular values as well as approximations to the Fourier coefficients directly from the CGLS iterations. This finding makes it possible to design a new stopping rule for the CGLS iterations, base upon the fact that the ratio between the Fourier coefficients and the singular values decays as long as we can extract information about the solution from the right-hand side.