VARIABLE-SEPARATION BASED ITERATIVE ENSEMBLE SMOOTHER FOR BAYESIAN INVERSE PROBLEMS IN ANOMALOUS DIFFUSION REACTION MODELS

The iterative ensemble smoother (IES) has been widely used to estimate parameters and states of dynamic models where the data are collected at all observation steps simultaneously. A large number of IES ensemble samples may be required in the estimation. This implies that we need to repeatedly compute the forward model corresponding to the ensemble samples. This leads to slow efficiency for large-scale and strongly nonlinear models. To accelerate the posterior inference in the estimation, a low rank approximation using a variable-separation (VS) method is presented to reduce the cost of computing the forward model. It will be efficient to construct a surrogate model based on the low rank approximation, which gives a separated representation of the solution for the stochastic partial differential equations (SPDEs). The separated representation is the product of deterministic basis functions and stochastic basis functions. For the anomalous diffusion reaction equations, the solution of the next moment depends on all of the previous moments, and this causes expensive computation for the Bayesian inverse problem. The presented VS can avoid this process through a few deterministic basis functions. The surrogate model can work well as the iteration moves on because the stochastic basis becomes more accurate when the uncertainty of random parameters decreases. To enhance the applicability in Bayesian inverse problems, we apply the VS-based IES method to complex structure patterns, which can be parameterized by discrete cosine transform (DCT). The post-processing technique based on a regularization method is employed after the iterations to improve the connectivity of the main features. In the paper, we focus on the time fractional diffusion reaction models in porous media and investigate their Bayesian inverse problems using the VS-based IES. A few numerical examples are presented to show the performance of the proposed IES method by taking account of structure inversion in permeability fields, parameters in permeability and reaction fields, and source functions.