Estimates of stability and propagation of errors in nonlinear inverse problems : with applications to delay-time tomography and inverse scattering transformations

Abstract

In order to apply nonlinear inversion methods to realistic data sets, effective regularization methods for nonlinear inverse problems have to be developed. It is remarkable that although significant progress has been made in the mathematical developments of truly nonlinear inverse problems, only regularization methods for weakly nonlinear inversion methods exist [13]. This makes that the application of truly nonlinear inversion methods in experimental research is still problematic. If nonlinear inversion methods are applied to real data, images of model functions can be obtained. It is however the question whether these model images are correct. Moreover, because of stability problems the question arises whether the images obtained from nonlinear inversions are really better than images obtained from a linearized inversion. In this thesis two aspects of the stability matter of inverse problems are investigated. The first aspect deals with the problem that using data sets contaminated with little noise can lead to large discrepancies in the reconstructed model. The second aspect deals with the nonlinear propagation of errors.