Stochastic asymptotical regularization for linear inverse problems

TL;DR

This paper introduces Stochastic Asymptotical Regularization (SAR), a method for uncertainty quantification in solving ill-posed linear inverse problems, demonstrating its optimal regularization properties and practical advantages over deterministic approaches.

Contribution

The paper develops SAR as a novel stochastic regularization method, providing theoretical proofs of its regularizing properties and convergence, along with numerical schemes and real-world applications.

Findings

SAR achieves mean-square convergence for ill-posed problems.

SAR provides uncertainty quantification, revealing hidden information.

Numerical experiments demonstrate SAR's accuracy and advantages.

Abstract

We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an optimal-order regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.