Stochastic asymptotical regularization for nonlinear ill-posed problems

TL;DR

This paper extends stochastic asymptotical regularization (SAR) to nonlinear ill-posed inverse problems, demonstrating its convergence, accuracy, and ability to quantify uncertainty, improve solutions, and identify multiple solutions.

Contribution

The paper develops the regularization theory of SAR for nonlinear inverse problems, establishing convergence and rate results, and showcasing its advantages over deterministic methods.

Findings

SAR achieves mean-square convergence for nonlinear problems.

SAR quantifies uncertainty and improves accuracy.

Numerical examples demonstrate SAR's effectiveness and advantages.

Abstract

Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this paper, we extend the regularization theory of SAR for nonlinear inverse problems. By combining techniques from classical regularization theory and stochastic analysis, we prove the regularizing properties of SAR with regard to mean-square convergence. The convergence rate results under the canonical sourcewise condition are also studied. Several numerical examples are used to show the accuracy and advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can quantify the uncertainty in error estimates for ill-posed problems, improve accuracy by selecting the optimal path, escape local minima for nonlinear problems, and identify multiple solutions by clustering samples of obtained approximate solutions.