Unscented Kalman Inversion

A useful approach to solve inverse problems is to pair the parameter-to-data map with a stochastic dynamical system for the parameter, and then employ techniques from filtering to estimate the parameter given the data. Three classical approaches to filtering of nonlinear systems are the extended, ensemble and unscented Kalman filters. The extended Kalman inversion (ExKI) is impractical when the forward map is not readily differentiable and given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Ensemble Kalman inversion (EKI) has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we demonstrate that unscented Kalman methods also provide an effective tool for derivative-free inversion in the setting of black-box forward models, introducing unscented Kalman inversion (UKI). Theoretical analysis is provided for linear inverse problems, and a smoothing property of the data mis-fit under the unscented transform is explained. We provide numerical experiments, including various applications: learning subsurface flow permeability parameters; learning the structure damage field; learning the Navier-Stokes initial condition; and learning subgrid-scale parameters in a general circulation model. The theory and experiments show that the UKI outperforms the EKI on parameter learning problems with moderate numbers of parameters and outperforms the ExKI on problems where the forward model is not readily differentiable, or where the derivative is very sensitive. In particular, UKI based methods are of particular value for parameter estimation problems in which the number of parameters is moderate but the forward model is expensive and provided as a black box which is impractical to differentiate.