文章基本信息

标题：Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems
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作者：Abhishake Rastogi ; Gilles Blanchard ; Peter Mathé 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2020
卷号：14
期号：2
页码：2798-2841
DOI：10.1214/20-EJS1735
语种：English
出版社：Institute of Mathematical Statistics
摘要：We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.
关键词：Statistical inverse problem;Tikhonov regularization;reproducing kernel Hilbert space;general source condition;minimax convergence rates