  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.
