Model Reduction Methods For Bayesian Inverse Problems In Partial Differential Equations

In this paper,we present some model reduction methods and their applications in inverse problems.We study the partial differential equation(PDE)involved inverse problems under the framework of the Bayesian inference,the reduced models accelerate the MCMC sampling process,which is used to explore the posterior.We also study the Kullback-Leibler(KL)divergence between the approximate posterior density and the reference one,to verify the performance of the reduced methods.We construct the surrogates using generalized polynomial chaos(gPC)technique,unlike the prior based polynomial chaos expansion(PCE),we use the coarse models constructed by the generalized Multiscale Finite Element Method(GMsFEM),to obtain an data informed intermediate distribution first,and then construct the PCE based on the intermediate distribution.We implement our strategy in fractional PDE inverse problems,the gPC surrogate constructed based on the intermediate distribution leads to better approximate posterior density than the one derived by the original prior.As the unimportant region of the posterior is excluded when construct the intermediate distribution,the acceptance rate of the Markov chains has been improved significantly.A low rank approximation is introduced to the solution of the corresponding forward problem and admits a variable separation form in terms of stochastic basis functions and physical basis functions.The calculation of stochastic basis functions is computationally predominant for the low rank expression.To significantly improve the efficiency of constructing the low rank approximation,we propose a bi-fidelity model reduction based on a novel variable separation method,where a low-fidelity model is used to compute the stochastic basis functions,and a highfidelity model is used to compute the physical basis functions.The low-fidelity model has lower accuracy but efficient to evaluate compared with the high-fidelity model,it accelerates the derivative of recursive formulation for the stochastic basis functions.The high-fidelity model is computed in parallel for a few samples scattered in the stochastic space when we construct the high-fidelity physical basis functions.The required number of forward model simulations in constructing the basis functions is very limited.The bi-fidelity model can be constructed efficiently while retaining a good accuracy simultaneously.In the proposed approach,both the stochastic basis functions and physical basis functions are calculated using the model information.This implies that a few basis functions may accurately represent the model solution in high-dimensional stochastic spaces.The bi-fidelity model reduction is applied to Bayesian inverse problems to accelerate the posterior exploration.A few numerical examples in time-fractional derivative diffusion models are carried out to identify smooth field and channel structured field in porous media in the framework of Bayesian inverse problems.When the dimension of unknown parameters is too high,such as when the inputs are fields represented with spatial discretizations of high dimension,the reduced methods introduced before will not work any more,an ensemble-based variable separation(VS)multiscale method is provided here.The ensemble-based 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 stochastic basis functions 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 KL divergence under the hybrid prior.