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Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Abhishake Rastogi

Abstract

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Abstract

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

Paper Structure

This paper contains 8 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. Reproducing Kernel Hilbert space and related operators
  4. The true solution, noise condition, and nonlinearity structure
  5. Effective dimension
  6. Convergence analysis
  7. Discussion
  8. Probabilistic estimates and preliminaries results

Key Result

Theorem 3.1

Let $\mathbf{z}$ be i.i.d. samples drawn according to the probability measure $\rho$. If Assumptions assmpt1--A.assumption and l.la.condition hold true and if $f_\rho -\bar{f}\in \mathcal{H}_q$ for some $q \in[1,2 + p]$. Then, for the Tikhonov estimator $f_{\mathbf{z},\lambda}~$ in (Tikhonov) with t

Theorems & Definitions (13)

  • Definition 2.1: Vector-valued reproducing kernel Hilbert space
  • Definition 2.2: Operator-valued positive semi-definite kernel
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Proposition A.1
  • Proposition A.2
  • Lemma A.3
  • ...and 3 more