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On saturation of the discrepancy principle for nonlinear Tikhonov regularization in Hilbert spaces

Qinian Jin

TL;DR

This paper analyzes the discrepancy principle for nonlinear Tikhonov regularization of ill-posed equations $F(x)=y$ in Hilbert spaces. It proves new saturation results under weaker conditions than previously known, by connecting the nonlinear discrepancy principle to a linearized problem with $A=F'(x^\dag)$ and leveraging spectral properties of $AA^*$; in particular, if the spectrum admits $\lambda_k\to 0$ and the worst-case error is $o(\delta^{1/2})$, then $x^\dag=x^*$. The results extend to a sequential discrepancy principle and establish sharp rate limits under Lipschitz continuity of $F'$, showing no improvement beyond $o(\delta^{1/2})$ unless $x^\dag=x^*$. Collectively, these findings clarify the stability and parameter-choice behavior of nonlinear regularization methods in Hilbert spaces, with implications for practical ill-posed problem solving.

Abstract

In this paper we revisit the discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems in Hilbert spaces and provide some new and improved saturation results under less restrictive conditions, comparing with the existing results in the literature.

On saturation of the discrepancy principle for nonlinear Tikhonov regularization in Hilbert spaces

TL;DR

This paper analyzes the discrepancy principle for nonlinear Tikhonov regularization of ill-posed equations in Hilbert spaces. It proves new saturation results under weaker conditions than previously known, by connecting the nonlinear discrepancy principle to a linearized problem with and leveraging spectral properties of ; in particular, if the spectrum admits and the worst-case error is , then . The results extend to a sequential discrepancy principle and establish sharp rate limits under Lipschitz continuity of , showing no improvement beyond unless . Collectively, these findings clarify the stability and parameter-choice behavior of nonlinear regularization methods in Hilbert spaces, with implications for practical ill-posed problem solving.

Abstract

In this paper we revisit the discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems in Hilbert spaces and provide some new and improved saturation results under less restrictive conditions, comparing with the existing results in the literature.
Paper Structure (2 sections, 4 theorems, 45 equations)

This paper contains 2 sections, 4 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1

Let Assumption Ass.1 and Assumption Ass.2 hold. Assume that $F'(x^\dag)$ is compact with infinite rank and $\max\{\kappa_0, \kappa_1\} \|x^*-x^\dag\|$ is sufficiently small. Consider Tikhonov regularization (TRH3) and let $\alpha(\delta, y^\delta)$ be determined by Rule Rule:DP with sufficiently lar as $\delta\to 0$, then $x^\dag = x^*$.

Theorems & Definitions (6)

  • Theorem 1: S1993a
  • Theorem 2
  • proof
  • Lemma 3
  • Theorem 4
  • proof