论文标题
非线性不良问题的随机渐近正规化
Stochastic asymptotical regularization for nonlinear ill-posed problems
论文作者
论文摘要
最近，在（\ emph {逆问题}，39：015007，2023）中开发了随机渐近正规化（SAR），用于对线性不良反问题的稳定近似解决方案的不确定性定量。在本文中，我们扩展了非线性反问题的SAR的正则化理论。通过结合经典正则化理论和随机分析的技术，我们证明了SAR在均方收敛方面的正则特性。还研究了在规范源条件下的收敛速率结果。使用几个数值示例来显示SAR的准确性和优势：与常规的确定性正规化方法相比，SAR可以量化缺陷问题的错误估计值的不确定性，通过选择最佳路径，逃脱非线性问题的局部最小值来提高准确性，以识别多个解决方案，并通过群集近似溶液来识别多个溶液。
Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this paper, we extend the regularization theory of SAR for nonlinear inverse problems. 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 quantify the uncertainty in error estimates for ill-posed problems, improve accuracy by selecting the optimal path, escape local minima for nonlinear problems, and identify multiple solutions by clustering samples of obtained approximate solutions.
