Nonlinear Tikhonov Regularization for Statistical Inverse Problems

• Activity Number: 51
• Type: Contributed
• Date/Time: Sunday, August 8, 2004 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Nicolai Bissantz*+
• Companies: University of Goettingen
• Keywords: statistical inverse problems ; nonlinear Tikhonov regularization ; nonparametric regression ; local polynomial estimators ; cross-validation

Abstract

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 L^2-function. The given data consist of measurements of u at n points, perturbed by random noise. We construct an estimator \hat{a}_n 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 (MISE) tends to 0 as n\to\infty under reasonable assumptions. Moreover, if a satisfies a source condition, we show for \hat a_n 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. Finally, the feasibility of our algorithm is investigated in a numerical study for a groundwater filtration problem and an inverse obstacle scattering problem.