  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.
