  Ensemble Kalman filter (EnKF) has been widely used in state estimation and
parameter estimation for the dynamic system where observational data is
obtained sequentially in time.
  To reduce uncertainty and accelerate posterior inference, a two-stage
ensemble Kalman filter is presented to improve the sequential analysis of EnKF.
It is known that the final posterior ensemble may be concentrated in a small
portion of the entire support of the initial prior ensemble. It will be much
more efficient if we first build a new prior by some partial observations, and
construct a surrogate only over the significant region of the new prior. To
this end, we construct a very coarse model using generalized multiscale finite
element method (GMsFEM) and generate a new prior ensemble in the first stage.
GMsFEM provides a set of hierarchical multiscale basis functions supported in
coarse blocks. This gives flexibility and adaptivity to choosing degree of
freedoms to construct a reduce model. In the second stage, we build an initial
surrogate model based on the new prior by using GMsFEM and sparse generalized
polynomial chaos (gPC)-based stochastic collocation methods. To improve the
initial surrogate model, we dynamically update the surrogate model, which is
adapted to the sequential availability of data and the updated analysis. The
two-stage EnKF can achieve a better estimation than standard EnKF, and
significantly improve the efficiency to update the ensemble analysis (posterior
exploration). To enhance the applicability and flexibility in Bayesian inverse
problems, we extend the two-stage EnKF to non-Gaussian models and hierarchical
models. In the paper, we focus on the time fractional diffusion-wave models in
porous media and investigate their Bayesian inverse problems using the proposed
two-stage EnKF.
