论文标题
线性反问题的随机渐近正规化
Stochastic asymptotical regularization for linear inverse problems
论文作者
论文摘要
我们介绍了随机渐近正规化（SAR）方法，用于对稳定的线性线性操作器方程的稳定近似解决方案进行不确定性定量，这是科学和工程中众多逆问题的确定性模型。我们证明了SAR在均方收敛方面的正则特性。我们还表明，只要根据溶液的平滑度选择SAR的终止时间，SAR是线性不良问题的最佳订购正则化方法。在一般范围类型的源条件下，先验和后部停止规则证明了这一结果。此外，还验证了SAR的一些相反结果。开发了两个迭代方案以实现SAR的数值实现，并提供了这两个数值方案的收敛分析。研究了一个玩具示例和一个现实世界中的生物传感器层析成像问题，以显示SAR的准确性和优势：与常规确定性逆问题的常规确定性正则方法相比，SAR可以提供不确定性量化的兴趣数量，从而可以通过散发出关于现实的数字来揭示和明确的信息，这些信息可以揭示和明确的信息，而这些信息是由噪音弥漫的，并且是愚蠢的，而散发出了不确定的噪音，并散发出了不确定性的模型，并且散发出了涉及的数字，并且是散发性的，并且是散发性的，并且是散发性的，并且是散发性的，而散发性的数字和散发性。
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.
