Abstract

We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed operator equations, which are deterministic models for numerous inverse problems in science and engineering. We demonstrate that SAR can quantify the uncertainty in error estimates for inverse problems. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an order-optimal 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. Some numerical examples are provided.