Abstract
We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. It is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution. For unknown smoothness the stopping index can be chosen adaptively based on Lepski's balancing principle. For this stopping rule we establish an oracle inequality, which implies order optimal rates for deterministic errors, and optimal rates up to a logarithmic factor for random noise. The performance and the statistical properties of the proposed method are illustrated by Monte Carlo simulations.
Highlights
In this paper we study the solution of nonlinear ill-posed operator equations
F (a) = u, assuming that the exact data u are perturbed by random noise
We will consider the case that the error in the data consists of both deterministic and stochastic parts, but we are mainly interested in the situation where the stochastic noise is dominant
Summary
In this paper we study the solution of nonlinear ill-posed operator equations (1.1). F (a) = u, assuming that the exact data u are perturbed by random noise. As the most common deterministic stopping rule, works reasonably well for discrete random noise models with small data vectors, the performance of the discrepancy principle becomes arbitrarily bad as the size of the data vector increases An important alternative to the iteratively regularized Gauss–Newton method is nonlinear Tikhonov regularization, for which convergence and convergence rate results for random noise have been obtained by O’Sullivan [23], Bissantz, Hohage, and Munk [4], Loubes and Ludena [17], and Hohage and Pricop [14]. The goal of the following analysis is to show that the nonlinearity error in terms of sharp estimates of Ekapp + Eknoi (Lemma 2.2)
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