  The ensemble Kalman inversion (EKI), as a derivative-free methodology, has
been widely used in the parameter estimation of inverse problems.
Unfortunately, its cost may become moderately large for systems described by
high dimensional nonlinear PDEs, as EKI requires a relatively large ensemble
size to guarantee its performance. In this paper, we propose an adaptive
multi-fidelity polynomial chaos (PC) based EKI technique to address this
challenge. Our new strategy combines a large number of low-order PC surrogate
model evaluations and a small number of high-fidelity forward model
evaluations, yielding a multi-fidelity approach. Especially, we present a new
approach that adaptively constructs and refines a multi-fidelity PC surrogate
during the EKI simulation. Since the forward model evaluations are only
required for updating the low-order multi-fidelity PC model, whose number can
be much smaller than the total ensemble size of the classic EKI, the entire
computational costs are thus significantly reduced. The new algorithm was
tested through the two-dimensional time fractional inverse diffusion problems
and demonstrated great effectiveness in comparison with PC based EKI and
classic EKI.
