  In this paper, we establish an initial theory regarding the stochastic
asymptotical regularization (SAR) for the uncertainty quantification of the
stable approximate solution of ill-posed nonlinear-operator equations, which
are deterministic models for numerous inverse problems in science and
engineering. 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 provide the uncertainty quantification of a solution and escape local
minimums for nonlinear problems.
