Ensemble-based implicit sampling for Bayesian inverse problems with non-Gaussian priors
For practitioners solving Bayesian inverse problems with non-Gaussian priors, this method improves applicability of implicit sampling by reducing computational cost, though it is an incremental extension of existing techniques.
The paper develops an ensemble-based implicit sampling method for Bayesian inverse problems that integrates iterative ensemble smoother (IES) with implicit sampling to handle non-Gaussian priors via discrete cosine transform and Gaussian mixture models, avoiding explicit Jacobian/Hessian computation. Numerical examples demonstrate efficacy for subsurface flow and anomalous diffusion models.
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.