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.

Short bio

张晔，深圳北理莫斯科大学和北京理工大学双聘教授、博士生导师，莫大-北理-深北莫计算数学与控制联合研究中心执行主任。国家高层次青年人才计划获得者(2020)、德国洪堡学者(2017)，2022年世界数学家大会Kovalevskaya奖获得者。2014年获得莫斯科国立大学数学物理副博士。主要研究领域是数学物理反问题的数学建模、数学理论和科学计算。在应用数学和统计学的国际顶级杂志发表高水平论文30多篇。目前主持国家重点研发青年科学家项目，北京市重点项目、国家自然科学基金面上项目、广东省和深圳市等多项省部级项目。