Na Ou

Zhao Zhang, Mengyao Xia, Na Ou

The uncertainty quantification (UQ) for mathematical models with random parameters is important for many science and engineering problems. Forward UQ quantifies the impact of random parameters on the output of system. In the current study, we propose a new stochastic finite volume (SFV) scheme by combining SFV with clustering algorithm in the parameter space such that each cluster can be regarded as a finite volume with implicit boundaries. The advantage of SFV is that no specific form of the random variable is required such that discontinuous solutions and sharp interfaces can be accurately approximated. Compared to classic SFV based on structured grids, the new SFV-cluster scheme extends SFV to parameter spaces of higher dimensions. For demonstration and validation, we present the construction of SFV schemes for the Kraichnan-Orszag three-mode problem and the Buckley-Leverett equation. The error analysis of SFV is extended and the new algorithm is validated by numerical experiments.

Lijian Jiang, Na Ou

This work presents a model reduction approach to the inverse problem in the application of subsurface flows. 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.

Na Ou, Guang Lin, Lijian Jiang

This work presents a multiscale model reduction approach to discontinuous fields identification problems in the framework of Bayesian inference. An ensemble-based variable separation (VS) method is proposed to approximate multiscale basis functions used to build a coarse model. The variable-separation expression is constructed for stochastic multiscale basis functions based on the random field, which is treated Gauss process as prior information. To this end, multiple local inhomogeneous Dirichlet boundary condition problems are required to be solved, and the ensemble-based method is used to obtain variable separation forms for the corresponding local functions. The local functions share the same interpolate rule for different physical basis functions in each coarse block. This approach significantly improves the efficiency of computation. We obtain the variable separation expression of multiscale basis functions, which can be used to the models with different boundary conditions and source terms, once the expression constructed. The proposed method is applied to discontinuous field identification problems where the hybrid of total variation and Gaussian (TG) densities are imposed as the penalty. We give a convergence analysis of the approximate posterior to the reference one with respect to the Kullback-Leibler (KL) divergence under the hybrid prior. The proposed method is applied to identify discontinuous structures in permeability fields. Two patterns of discontinuous structures are considered in numerical examples: separated blocks and nested blocks.

Lijian Jiang, Na Ou

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$_\text{ZS}$ 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.