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In the paper, we present a strategy for accelerating posterior inference for unknown inputs in time fractional diffusion models. In many inference problems, the posterior may be concentrated in a small portion of the entire prior support. It will be much more efficient if we build and simulate a surrogate only over the significant region of the posterior. To this end, we construct a coarse model using Generalized Multiscale Finite Element Method (GMsFEM), and solve a least-squares problem for the coarse model with a regularizing Levenberg-Marquart algorithm. An intermediate distribution is built based on the approximate sampling distribution. For Bayesian inference, we use GMsFEM and least-squares stochastic collocation method to obtain a reduced coarse model based on the intermediate distribution. To increase the sampling speed of Markov chain Monte Carlo, the DREAM ZS algorithm is used to explore the surrogate posterior density, which is based on the surrogate likelihood and the intermediate distribution. The proposed method with lower gPC order gives the approximate posterior as accurate as the the surrogate model directly based on the original prior. A few numerical examples for time fractional diffusion equations are carried out to demonstrate the performance of the proposed method with applications of the Bayesian inversion. ).Here the α is the regularization parameter and I is the identity matrix of size n z × n z , J is the sensitivity matrix, whose transpose is defined byThe gradient of G ri (z k ), i = 1, · · · , n d can be approximated by a difference method, e.g., each column of J can be computed by J(:, j) = G r (z k + hη j ) − G r (z k ) h

This work presents a model reduction approach to the inverse problem in the application of subsurface flows. One such an application is to estimate model's inputs and identify model's parameters. This is often challenging because the complicated multiscale structures are inherently in the model and the estimated inputs are parameterized in a high-dimensional space. We often need to estimate the probabilistic distribution of the unknown inputs based on some observations. Bayesian inference is desirable for this situation and solving the inverse problem. For the Bayesian inverse problem, the forward model needs to be repeatedly computed for a large number of samples to get a stationary chain. This requires large computational efforts. To significantly improve the computation efficiency, we use generalized multiscale finite element method and least-squares stochastic collocation method to construct a reduced computational model. To avoid the difficulty of choosing regularization parameter, hyperparameters are introduced to build a hierarchical model. We use truncated Karhunen-Loeve expansion (KLE) to reduce the dimension of the parameter spaces and decrease the mixed time of Markov chains. The techniques of hyperparameter and KLE are incorporated into the model reduction method. The reduced model is constructed offline. Then it is computed very efficiently in the online sampling stage. This strategy can significantly accelerate the evaluation of the Markov chain and the resultant posterior distribution converges fast. We analyze the convergence for the approximation between the posterior distribution by the reduced model and the reference posterior distribution by the full-order model. A few numerical examples in subsurface flows are carried out to demonstrate the performance of the presented model reduction method with application of the Bayesian inverse problem.

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. Very burdened simulations for the forward problem are needed to update a large number of EnKF ensemble samples. This will slow down the analysis efficiency of the EnKF for largescale and high dimensional models. 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. A few numerical examples are carried out to demonstrate the performance of the two-stage EnKF method by taking account of parameter and structure inversion in permeability fields and source functions. IntroductionThe model inputs (parameters, source, domain geometry and system structure, et. al.) in many practical systems are often unknown. We need to identify or estimate these inputs by partial and noisy observations to construct predictive models and calibrate the models. This results in inverse problems. In the paper, we consider the inverse problems in anomalous diffusion models. The anomalous diffusion can be roughly classified into two categories: subdiffusion (0 < γ < 1) and superdiffusion (1 < γ < 2). Here γ denotes the fraction derivative with respect to time. The anomalous diffusion equations are also called fractional diffusion-wave equations. Theoretical results such as existence, uniqueness, stability and numerical error estimates are presented in [19] for some type of anomalous diffusion equations. The rela...

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