We propose a data-assisted two-stage method for solving an inverse random source problem of the Helmholtz equation. In the first stage, the regularized Kaczmarz method is employed to generate initial approximations of the mean and variance based on the mild solution of the stochastic Helmholtz equation. A dataset is then obtained by sampling the approximate and corresponding true profiles from a certain a-priori criterion. The second stage is formulated as an image-to-image translation problem, and several data-assisted approaches are utilized to handle the dataset and obtain enhanced reconstructions. Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.

翻译：我们提出了一种数据辅助的两阶段方法，用于求解亥姆霍兹方程的逆随机源问题。在第一阶段，基于随机亥姆霍兹方程的温和解，采用正则化Kaczmarz方法生成均值和方差的初始近似值。随后，根据某种先验准则，通过采样近似及相应的真实剖面获得数据集。第二阶段被表述为图像到图像的翻译问题，并利用多种数据辅助方法处理该数据集以实现增强重建。数值实验表明，该数据辅助两阶段方法能够在更少实现次数的情况下，为均匀和非均匀介质提供令人满意的重建效果。