  In this study, we consider an ensemble Kalman inversion (EKI) for the
numerical solution of time-fractional diffusion inverse problems (TFDIPs).
Computational challenges in the EKI arise from the need for repeated
evaluations of the forward model. We address this challenge by introducing a
non-intrusive reduced basis (RB) method for constructing surrogate models to
reduce computational cost. In this method, a reduced basis is extracted from a
set of full-order snapshots by the proper orthogonal decomposition (POD), and a
doubly stochastic radial basis function (DSRBF) is used to learn the projection
coefficients. The DSRBF is carried out in the offline stage with a stochastic
leave-one-out cross-validation algorithm to select the shape parameter, and the
outputs for new parameter values can be obtained rapidly during the online
stage. Due to the complete decoupling of the offline and online stages, the
proposed non-intrusive RB method -- referred to as POD-DSRBF -- provides a
powerful tool to accelerate the EKI approach for TFDIPs. We demonstrate the
practical performance of the proposed strategies through two nonlinear
time-fractional diffusion inverse problems. The numerical results indicate that
the new algorithm can achieve significant computational gains without
sacrificing accuracy.
