Ensemble Methods For Bayesian Uncertainty Quantification Research

Bayesian uncertainty quantification has important applications in the mathematical modeling of complex systems.In inverse problems,most models have uncertainty.The classic Markov Chain Monte Carlo method is the basic method of following researches on Bayesian inverse problems,but this method has a slow convergence in high-dimensional space and the samples are correlated.Therefore,how to accelerate to explore the posterior distribution and obtain an accuracy estimate is the hot topic among scholars.This dissertation conducts research on ensemble methods for Bayesian uncertainty quantification,and studies the estimate of the posterior for the model’s inputs(parameters,source,domain geometry and system structure,et al.)of the strongly nonlinear porous media in the highdimensional space.The adopted ensemble method provides us with more efficiency and accuracy inference results than the classic statistical method.Firstly,we provide a two-stage ensemble Kalman filter based on multiscale model reduction method.Ensemble Kalman filter(EnKF)has been widely used in the state and parameter estimations of dynamical systems where observation data are sequentially obtained in time.Very burdened simulations for the forward problem are needed to update a large number of ensemble samples.This will slow down the analysis efficiency of the EnKF for large-scale and high dimensional models.Besides,the posterior distributions of most Bayesian inverse problems may be concentrated in a small portion of the entire support of the initial prior.Thus we provide a two-stage EnKF based on multiscale model reduction method.For the proposed method,the key is to construct a new prior and then build an efficiency surrogate model based on the new prior.Here the generalized multiscale finite element method(GMsFEM)can provide a set of hierarchical multiscale basis functions,which gives flexibility and adaptivity to choosing degree of freedoms to construct a reduce model.Moreover,we adopt the fixed point iteration method to construct the sparse generalized polynomial chaos based surrogate model.Due to the sequential availability of observations and the update analysis,the surrogate model is updated sequentially.Compared with the classic EnKF,numerical examples demonstrate that the proposed method not only provides a more accuracy and efficiency estimation,but also improve the flexibility and availability for Bayesian inverse problems.Secondly,based on the existing research of the mixture models,we conduct an extending research on a residual-driven adaptive Gaussian mixture approximation(RD-AGMA)method.The classical ensemble methods can provide a good approximation for the Gaussian or approximate Gaussian models.But they perform poorly for the multimodal posterior distributions.Thus we use Gaussian mixture model(GMM)to approximate the posterior in the multimodal models.To obtain an adaptive Gaussian mixture approximation(GMA),we select the mixture components based on the residual between the observation data and model response.To accelerate the computation efficiency,we apply a small sample size to compute all the covariance matrices.Then we get a GMA,which can generate a large number of samples without solving the forward model.A few numerical examples illustrate the efficiency of the proposed method with applications in multimodal inverse problems.Finally,based on the existing research of variable-separation(VS)and EnKF methods,we give a two-stage variable-separation Kalman filter(T-VSKF)method for data assimilation.The proposed method provides the VS expressions of the parameter and state based on a novel VS method.Here we only need to update the coefficients of the VS expression.Due to the effectiveness of the VS representation,the number of VS terms is not the exponential increasing as the dimension of unknowns increases.This avoids the dimension curse.In the T-VSKF method,the first stage is to locate an approximate mean,and the second stage is to improve the mean and reduce the uncertainty of the unknown parameters and state.A few numerical examples are presented to show the efficacy of T-VSKF in the data assimilation.Extensive comparison is made for the proposed method and another three filter methods(EnKF,ensemble square-root filter(EnSRF)and VSKF)in these numerical examples.The numerical results demonstrate that T-VSKF achieves the best accuracy and efficiency among the four methods.