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

TL;DR

This paper develops a new approach to derive convergence rates for Landweber-type methods in Banach spaces, accommodating variable step sizes and stochastic variants, advancing understanding of their efficiency in solving ill-posed inverse problems.

Contribution

It introduces a novel strategy for obtaining convergence rates under a benchmark source condition, applicable to both deterministic and stochastic Landweber-type methods with flexible step sizes.

Findings

Derived convergence rates under a benchmark source condition.

Extended the approach to stochastic mirror descent methods.

Achieved almost sure convergence rate in stochastic settings.

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.