A new Approach to the Solution of Linear and Nonlinear Inverse Problems: Regularization for Curve and Surface Representations

Many relevant problems in, practice are so-called inverse problems, i.e., problems concerned with determining causes for a desired or an observed effect. Besides,some linear problems, like, e.g., deblurring problems in signal and image processing or denoising problems, frequently many nonlinear problems arise in applications in science and technology, e.g., in parameter identification. The mathematical formulation of these problem usually gives rise to ill-posed problems, i.e., their solution is unstable under data perturbations. Whereas the mathematical theory of linear ill-posed problems has been extensively studied there are still many open questions in the nonlinear case. In many practical applications, particularly in signal and image processing and parameter identification problems, one is interested in discontinuous solutions of the appropriate equations. Via regularization methods typically stability is obtained by imposing smoothness constraints on the approximate solutions. A serious shortcoming of standard regularization techniques is that they do not yield good results for discontinuous solutions. A regularization technique which turned out to yield more satisfactory results is bounded variation regularization. However, the numerical realization of this method is usually done by modifying the bounded variation norm. These modifications are counterproductive to the original idea of efficiently locating discontinuities. Moreover, convergence of bounded variation regularized solutions can be only guaranteed in some Lp -norm. Recently, a new approach has been developed by the proposer (and O. Scherzer) that allows reconstructions which approximate a discontinuous solution in a uniform sense. In this approach, a function is interpreted as a parameterized curve. In this project we want to extend this approach to nonlinear inverse problems and especially to two-dimensional problems. There the following questions have to be attacked: How should the representation be done by surfaces? What regions of discontinuities can be efficiently realized by fast numerical algorithms? How fast will the convergence of the regularized solutions be near and away from discontinuities? The project will involve both, theoretical research and intensive numerical computation.