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

摘要

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 L-2-function. The given data consist of measurements of u at n points, perturbed by random noise. We construct an estimator (a) over cap (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 Eparallel to(a) over cap (n) - aparallel to(H)(2) (MISE) tends to 0 as n --> infinity under reasonable assumptions. Moreover, if a satisfies a source condition, we prove convergence rates for the MISE of (a) over cap (n) 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.