Newton type methods for nonlinear ill-posed problems with applications to nondestructive testing

Nondestructive testing of material plays a major role in many scientific, medical, and industrial applications. Its mathematical formulation leads to a nonlinear inverse problem, in the abstract formulation F(x) = y, where F is a nonlinear operator between Banach spaces X and Y. These inverse problems are typically ill-posed in the sense of lack of stable data dependence of a solution. Numerical methods tackling this instability are called regularization methods, probably the most well-known among them being Tikhonov regularization. Recently, also iterative regularization methods have been developped and analyzed, among them especially stabilized versions of the Newton iteration xn+1 + = xn - F`(xn ) -1 F(x n ), whose excellent local convergence properties in the well-posed situation are well known and could in part be carried over also to the ill-posed setting. Due to the nonlinearity of the underlying problem only local convergence of these iterations could be shown. It is, of course, desirable to have convergence from an arbitrary starting point, therefore a main aim of this project is to develop globalization strategies for iterative regularization methods for nonliear ill-posed problems on the basis of linesearch, trust region and continuation methods, that have already been successfully applied for globalizing solution methods for nonlinear well-posed problems. Another task of this project is to incorporate constraints on x, as they naturally arise, e.g. from the physical background of the concrete inverse problem under consideration, in the known iterative regularization methods for nonlinear problems. Furthermore the developped algorithms are supposed to be applied in numerical simulations of nondestructive testing problems, especially in bond strength measurements by ultrasonics. Here a strong cooperation with the Institute of Electrical Measurement Technique in Linz would be possible.