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Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with Hölder Stability Estimate

Akari Ishida, Sei Nagayasu, Gen Nakamura

Abstract

We analyze convergence of the Levenberg-Marquardt method for solving nonlinear inverse problems in Hilbert spaces. Specifically, we establish local convergence and convergence rates for a class of inverse problems that satisfy Hölder stability estimate. Furthermore, based on what we found in the mentioned analysis, we develop global reconstruction algorithms for solving inverse problems with finite measurements for exact and noisy data, respectively.

Convergence Analysis of Levenberg-Marquardt Method for Inverse Problem with Hölder Stability Estimate

Abstract

We analyze convergence of the Levenberg-Marquardt method for solving nonlinear inverse problems in Hilbert spaces. Specifically, we establish local convergence and convergence rates for a class of inverse problems that satisfy Hölder stability estimate. Furthermore, based on what we found in the mentioned analysis, we develop global reconstruction algorithms for solving inverse problems with finite measurements for exact and noisy data, respectively.
Paper Structure (6 sections, 7 theorems, 98 equations)

This paper contains 6 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence for exact data
  4. Analysis for noisy data
  5. Application to IFM
  6. Summary and discussion

Key Result

Lemma 2.1

Let $0<q<1$. We assume that unique;alpha holds so that $\alpha_{k}$ can be defined via MDP. We define $x_{k}^{\delta}$ as LM. Then, we have

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Lemma 2.2: monotonicity of error, KNS
  • proof
  • Lemma 2.3
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 7 more