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Convergence rates of Landweber-type methods for inverse problems in Banach spaces

Qinian Jin

Abstract

Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper, we tackle the challenge of deriving convergence rates for Landweber-type methods applied to ill-posed inverse problems, where forward operators map from a Banach space to a Hilbert space. Under a benchmark source condition, we introduce a novel strategy to derive convergence rates when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. Our results offer substantial flexibility regarding step sizes, by allowing the use of variable step sizes. By extending the strategy to deal with the stochastic mirror descent method for solving nonlinear ill-posed systems with exact data, under a benchmark source condition we also obtain an almost sure convergence rate in terms of the number of iterations.

Convergence rates of Landweber-type methods for inverse problems in Banach spaces

Abstract

Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper, we tackle the challenge of deriving convergence rates for Landweber-type methods applied to ill-posed inverse problems, where forward operators map from a Banach space to a Hilbert space. Under a benchmark source condition, we introduce a novel strategy to derive convergence rates when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. Our results offer substantial flexibility regarding step sizes, by allowing the use of variable step sizes. By extending the strategy to deal with the stochastic mirror descent method for solving nonlinear ill-posed systems with exact data, under a benchmark source condition we also obtain an almost sure convergence rate in terms of the number of iterations.

Paper Structure

This paper contains 8 sections, 14 theorems, 148 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence rates of Landweber-type methods
  4. Some key estimates
  5. Convergence rate under a priori stopping rule
  6. Convergence rate under the discrepancy principle
  7. Convergence rate of stochastic mirror descent method
  8. Numerical results

Key Result

Theorem 3.1

Let Assumption ass0 and Assumption ass1 hold. Consider the Landweber-type method (Land). Let $\tau >(1+\eta)/(1-\eta)$ be a given number and let $\{\gamma_k^\delta\}$ be chosen by one of the following rules: Then the discrepancy principle (DP0) outputs a finite integer $k_\delta$ and there is a solution $x^*$ of (Land.eq) in $B_{2\rho}(x_0) \cap \emph{dom}({\mathcal{R}})$ such that If in additio

Theorems & Definitions (32)

  • Theorem 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • Theorem 3.8
  • ...and 22 more