  In the paper, we develop an ensemble-based implicit sampling method for
Bayesian inverse problems. For Bayesian inference, the iterative ensemble
smoother (IES) and implicit sampling are integrated to obtain importance
ensemble samples, which build an importance density. The proposed method shares
a similar idea to importance sampling. IES is used to approximate mean and
covariance of a posterior distribution. This provides the MAP point and the
inverse of Hessian matrix, which are necessary to construct the implicit map in
implicit sampling. The importance samples are generated by the implicit map and
the corresponding weights are the ratio between the importance density and
posterior density. In the proposed method, we use the ensemble samples of IES
to find the optimization solution of likelihood function and the inverse of
Hessian matrix. This approach avoids the explicit computation for Jacobian
matrix and Hessian matrix, which are very computationally expensive in high
dimension spaces. To treat non-Gaussian models, discrete cosine transform and
Gaussian mixture model are used to characterize the non-Gaussian priors. The
ensemble-based implicit sampling method is extended to the non-Gaussian priors
for exploring the posterior of unknowns in inverse problems. The proposed
method is used for each individual Gaussian model in the Gaussian mixture
model. The proposed approach substantially improves the applicability of
implicit sampling method. A few numerical examples are presented to demonstrate
the efficacy of the proposed method with applications of inverse problems for
subsurface flow problems and anomalous diffusion models in porous media.
