Stochastic mirror descent method for linear ill-posed problems in Banach spaces

TL;DR

This paper introduces a stochastic mirror descent method for large-scale linear ill-posed problems in Banach spaces, which efficiently uses random subsets of equations and regularization to improve scalability and solution quality.

Contribution

The paper develops a novel stochastic mirror descent algorithm tailored for large-scale ill-posed problems, incorporating convex regularization and adaptive step-size rules for enhanced convergence.

Findings

Method scales well with problem size

Incorporates regularization to improve solution stability

Achieves order optimal convergence under source conditions

Abstract

Consider linear ill-posed problems governed by the system $A_i x = y_i$ for $i =1, \cdots, p$, where each $A_i$ is a bounded linear operator from a Banach space $X$ to a Hilbert space $Y_i$. In case $p$ is huge, solving the problem by an iterative regularization method using the whole information at each iteration step can be very expensive, due to the huge amount of memory and excessive computational load per iteration. To solve such large-scale ill-posed systems efficiently, we develop in this paper a stochastic mirror descent method which uses only a small portion of equations randomly selected at each iteration steps and incorporates convex regularization terms into the algorithm design. Therefore, our method scales very well with the problem size and has the capability of capturing features of sought solutions. The convergence property of the method depends crucially on the choice of step-sizes. We consider various rules for choosing step-sizes and obtain convergence results under {\it a priori} early stopping rules. In particular, by incorporating the spirit of the discrepancy principle we propose a choice rule of step-sizes which can efficiently suppress the oscillations in iterates and reduce the effect of semi-convergence. Furthermore, we establish an order optimal convergence rate result when the sought solution satisfies a benchmark source condition. Various numerical simulations are reported to test the performance of the method.