Berlin 2005 – scientific programme
Parts | Days | Selection | Search | Downloads | Help
DY: Dynamik und Statistische Physik
DY 40: Statistical Physics (General) I
DY 40.5: Talk
Tuesday, March 8, 2005, 11:15–11:30, TU H3010
Nonlinear Tikhonov regularization for inverse problems with random noise — •Nicolai Bissantz1, Thorsten Hohage2, and Axel Munk1 — 1Institut für Mathematische Stochastik der Universität Göttingen — 2Institut für Numerische und Angewandte Mathematik der Universität Göttingen
We consider nonlinear statistical inverse problems described by operator equations F(a)=u. Here a is an element of a Hilbert space which we want to estimate, and u is an L2-function. The given data consist of measurements of u at n points, perturbed by random noise. We construct an estimator ân for a by a combination of a local polynomial estimator and a nonlinear Tikhonov regularization and establish consistency in the sense that the mean integrated square error E|ân−a|2 (MISE) tends to 0 as n→∞ under reasonable assumptions. Moreover, if a satisfies a source condition, we show for ân a convergence rate result for the MISE, as well as almost surely. Further, it is shown that a cross validated parameter selection yields a fully data driven consistent method for the reconstruction of a. Finally, the feasibility of our algorithm is investigated in a numerical study for a groundwater filtration problem and an inverse obstacle scattering problem.