The unscented Kalman inversion (UKI) method presented in [1] is a general derivative-free approach for the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be differentiable. The regularization strategies, convergence property, and speed-up strategies [1,2] of the UKI are thoroughly studied, and the method is capable of handling noisy observation data and solving chaotic inverse problems. In this paper, we study the uncertainty quantification capability of the UKI. We propose a modified UKI, which allows to well approximate the mean and covariance of the posterior distribution for well-posed inverse problems with large observation data. Theoretical guarantees for both linear and nonlinear inverse problems are presented. Numerical results, including learning of permeability parameters in subsurface flow and of the Navier-Stokes initial condition from solution data at positive times are presented. The results obtained by the UKI require only $O(10)$ iterations, and match well with the expected results obtained by the Markov Chain Monte Carlo method.

翻译：[1] 中提出的非中度 Kalman 倒置法(UKI) 方法在[1] 中是针对反问题的一种一般无衍生物处理方法。 UKI 特别适合反向问题,因为前方模型是黑盒,可能无法区分。 UKI 的正规化战略、趋同属性和加速战略[1,2] 得到了彻底研究,该方法能够处理噪音观测数据和解决反向混乱问题。在本文件中,我们研究了UKI 的不确定性量化能力。我们提出了修改的UCI,它使得远端模型分布的平均值和共性能够与大度观测数据的反差问题相近。线性和非线性反性问题的理论保障得到了介绍。数字结果,包括在积极时间从解决方案数据中学习了地表下流的渗透参数和Navier-Stoks的初步条件。UKI 获得的结果只需要$(10) 美元 Iteration,并与Markov Cain Conte Carlo 方法的预期结果相匹配。