Stochastic gradient descent method with convex penalty for ill-posed problems in Banach spaces

TL;DR

This paper proposes a stochastic gradient descent method with convex penalties for solving ill-posed inverse problems in Banach spaces, featuring adaptive step size and regularization strategies, validated through numerical experiments.

Contribution

It introduces a scalable SGD method with convex penalties and adaptive step size for ill-posed problems in Banach spaces, including regularization and stopping rules.

Findings

Effective in computed tomography and schlieren imaging

Demonstrates regularization properties under certain conditions

Shows finite iteration termination with an a posteriori rule

Abstract

In this work, we investigate a stochastic gradient descent method for solving inverse problems that can be written as systems of linear or nonlinear ill-posed equations in Banach spaces. The method uses only a randomly selected equation at each iteration and employs the convex function as the penalty term, and thus it is scalable to the problem size and has the ability to detect special features of solutions such as nonnegativity and piecewise constancy. To suppress the oscillation in iterates and reduce the semi-convergence of such methods, by incorporating the spirit of discrepancy principle, an adaptive strategy for choosing the step size is suggested. Under certain conditions, we establish the regularization results of the method under an {\it a priori} stopping rule. Several numerical simulations on computed tomography and schlieren imaging are provided to demonstrate the effectiveness of the method. Finally, we study an {\it a posteriori} stopping rule for SGD-$\theta$ method and show the finite iterations termination property.