Linear, Non-linear, Non-differentiable, Non-convex regularization involving Unbounded Operators

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

F(x)=y

by a minimizer of the penalized functional

||F(x)-y||^2 +\alpha ||x||^2 (\alpha >0) .

In the beginning mainly linear ill-posed problems (i.e. F 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, Engl & Groetsch, Groetsch, Morozov, Louis Natterer, Bertero & Boccacci, Kirsch, Colton & Kress... . In 1989 Engl & Kunisch & Neubauer and Seidman & Vogel developed a regularization theory for non-linear inverse problems where F 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

||F(x)-y||^2 + \alpha\int |\nabla x| .

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

\int g(F(x)-y,x,\nabla x),

which may be nonconvex with respect to the third component \nabla x.

Another complication is introduced in the analysis of regularization functionals if for instance the operator F can be decomposed into a continuous and a discontinuous operator. Such models have become popular for level set regularization recently.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).