2003-2004

PROFESSOR OTMAR SCHERZER

Department of Computer Science

University of Innsbruck, Austria

will give a talk on

Regularization for ill-posed problems: Linear gNon-linear g

Non-differentiable g Non-convex

Friday, April 30, 2004

In 1963 Tikhonov initiated the research on stable method for the numerical solution of inverse and ill-posed problems. Tikhonov’s approach consists in approximating a solution of an operator equation

by a minimizer of the penalized functional

In the beginning mainly linear ill-posed problems (i.e. is linear) such as computerized tomography have been solved with these methods. The theory of Tikhonov regularization methods developed systematically. Until around 1980 there has been success in a rigorous and rather complete analysis of regularization methods for linear ill-posed problems. We mention the books of Tikhonov & Arsenin, Nashed, Groetsch, Morozov, Louis, Natterer, Bertero & Boccacci, Kirsch, Colton & Kress... In 1989 Engl, Hanke and Neubauer, Kunisch & Neubauer and Seidman & Vogel developed a regularization theory for non-linear inverse problems where is a non-linear, differentiable operator. About the same time Osher & Rudin used bounded variation regularization for denoising and deblurring, which consists in minimization of the functional

This method is highly successful in restoring discontinuities. The analysis of bounded variation regularization is significantly more involved since the penalization functional is not differentiable. Over the past years this concept has attracted many mathematical research. The next step toward generalization of regularization methods is non-convex regularization. Here the general goal is to minimize functionals of the form

which may be nonconvex with respect to the third component. For instance, Christensen has used such models for brain imaging. For the analysis we observe a new complication due to the nonconvexity: the generalized solution concepts of -limits and quasi-convexification from the calculus of variations have to be involved. In this talk we give an overview on the analysis of regularization models which is supported by numerical experiments.