Model Reduction Methods For High Dimensional Stochastic Problems And Its Application

One of the challenges of the high-precision numerical simulation of stochastic models with multiscale information is the reduction of the high-dimensional stochastic parameters.In order to overcome the difficulties brought by the multiscale information and the high-dimensional stochastic parameters effectively,we focuses on the multiscale method and the reduction of the high-dimensional random space.In the generalized Polynomial Chaos(gPC) methods,the coefficients provided the contribution of each random variable to the variance of local “snapshot” substitution model by the Variance Analysis-High Dimensional Model Representation(ANOVA-HDMR).Therefore,we propose a partitioning method of random space,which transforms the original problem of a large scale substitution model into a group of substitution problems of multiple random subdomains.Moreover,we combine with the random space partition method and the reduction method of MsFEM,propose a reduction method of Multielement Least Square-High Dimensional Model Representation(MeRLS-HDMR),including the moments relations between the local substitution model and the global substitution model.The expected,variance and statistical distribution of the water cut production with the change of Pore Volume of Injection(PVI) are given in the application experiment of the quarter five-spot problem of underground oil-water two-phase flow model.The results show that the Me RLS-HDRM method is effective under the above reduction strategies in a random space.Firstly,since the reduction performance of the local substitution model mainly depends on the reduction performance of the viscosity coefficient,the comparison of the reduction performance of the viscosity coefficient represented by KL basis and Fourier basis is studied.Under the condition of the same accuracy,the larger the correlation length of the Gaussian process representing the viscosity coefficient with the exponential kernel,the more sparse the basis of the solution.Conversely,if and only if the correlation length is sufficiently small,the result similar to KL lemma can be obtained.Secondly,inspired by the Random Walk(RW) method for the homogeneous problems without the source term,a numerical algorithm based on Markov chain is proposed for heterogeneous problems with a source term.The one-step transition probability matrix of Phase-type distribution is constructed by treating the internal nodes and the boundary nodes as the recurrent state and absorption state respectively.In order to calculate the probability of stop time within finite step of Phase type distribution,a method(EigRW) of eigenvalue decomposition of one step transition matrix and a method(Inv-RW) of inverse matrix of the stationary matrix within finite step are given.We prove the convergence of the Inv-RW algorithms using the Gerschgorin’s disk theorem effectively,and find that the convergence rate decreases exponentially with the increase of samples.It should be noted that it is different from the general Uncertainty Quantification(UQ) methods.Finally,based on the Multiscale Finite Element(MsFEM) and Generalized Multiscale Finite Element(GMsFEM) methods,the performance comparison of the Finite Element Method(FEM) and MsFEM/GMsFEM is studied in the thesis.In order to achieve the requirement of lower sparsity of the multiscale basis functions,a model reduction method(SVD-sGMsFEM) of the physical space and the random space is established respectively by singular value decomposition,and a comparative numerical experiments of MsFEM and GMsFEM for high contrast media is presented.The results show that MsFEM is significantly better than the classical FEM on coarse mesh in terms of accuracy,while GMsFEM is significantly better than MsFEM.