Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise

Abstract

We consider nonlinear inverse problems described by operator equations F(a) = u. Here a is an element of a Hilbert space H 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 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 → ∞ under reasonable assumptions. Moreover, if a satisfies a source condition, we prove convergence rates for the MISE of , 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, respectively.