Multiscale model reduction method for Bayesian inverse problems of subsurface flow

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 inherent 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 samplers 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.