The authors consider a randomized solution to ill-posed operator equations in Hilbert spaces. In contrast to statistical inverse problems, where randomness appears in the noise, here randomness arises in the low-rank matrix approximation of the forward operator, which results in using a Monte Carlo method to solve the inverse problems. In particular, this approach follows the paradigm of the study N. Halko et al 2011 SIAM Rev . 53 217–288, and hence regularization is performed based on the low-rank matrix approximation. Error bounds for the mean error are obtained which take into account solution smoothness and the inherent noise level. Based on the structure of the error decomposition the authors propose a novel algorithm which guarantees (on the mean) a prescribed error tolerance. Numerical simulations confirm the theoretical findings.

中文翻译：

---

随机矩阵逼近可增强逆问题中的正则投影方案

作者考虑了希尔伯特空间中不适定算子方程的随机解。与统计逆问题相反，在噪声中出现随机性，此处的随机性出现在前向算子的低秩矩阵逼近中，这导致使用蒙特卡洛方法来解决逆问题。特别地，这种方法遵循研究N. Halko等人2011 SIAM Rev.的范例。53 217–288，因此基于低秩矩阵逼近执行正则化。获得了平均误差的误差范围，其中考虑了解决方案的平滑度和固有噪声水平。基于误差分解的结构，作者提出了一种新颖的算法，该算法可以保证（平均）规定的误差容限。数值模拟证实了理论发现。