Model Reduction Methods For Stochastic Partial Differential Equations

In this doctoral dissertation,we mainly study model reduction methods for stochastic partial differential equations to improve the efficiency of numerical sim-ulation.In the framework of Galerkin projection,dimension reduction techniques are used to construct a reduced order model.We attempt to reduce the computa-tion cost as much as possible with a little loss of the accuracy.The variable-separation representation of the model's input?e.g.,coefficients and source terms?is crucial to achieve the decomposition of offline-online compu-tation.The separated representation can significantly improve the online compu-tation efficiency,which is helpful for estimating the influence of the uncertainty in-formation and improving the computation efficiency of the related inverse problem.In order to efficiently construct the separated representation of the model's inputs and outputs,we first introduce two strategies:Least-squares method of snapshots?LSMOS?and Sparse tensor approximation using orthogonal matching pursuit?S-TAOMP?.Let G?x,??be a generic function parameterized by ?.We want to find an approximation of G?x,??in S_Nsuch that G?x,???G_N?x,??:=?_i=1^N?_i???g_i?x?in a finite dimensional space.Let?:=??₁,···,?_n?be a set of n real-valued ran-dom variables,Then we present a sparse low rank tensor approximations?SLRA?method to construct the approximation for the multivariate function????in high stochastic dimension spaces.We try to get the sparse low rank tensor approxima-tion of????in the finite dimensional tensor space by successively computing sparse rank-one approximation.Secondly,we propose a novel variable-separation?NVS?method.We firstly introduce the NVS method for generic multivariate functions,which can be used to get the separated representation of model's input.The idea of the novel VS is extended to obtain the solution in tensor product structure for stochastic par-tial differential equations?SPDEs?.Compared with many widely used variation-separation methods,the novel VS shares their merits but has less computation complexity and better efficiency.The novel VS can be used to get the separat-ed representation of the solution for SPDE in a systematic enrichment manner.Saddle point problems often arise in a variety of applications in science and en-gineering.For example,mixed finite element methods in engineering application such as fluid and solid mechanics are the typical examples of saddle point systems and quadratic programming in optimal control is another popular application and so on.In order to effectively simulate stochastic saddle point problems,we con-sider a variable-separation?VS?method to solve the stochastic saddle point?SSP?problems.The VS method is applied to obtain the solution in tensor product structure for stochastic partial differential equations?SPDEs?in a mixed formula-tion.The aim of such a technique is to construct a reduced basis approximation of the solution of the SSP problems.The VS method attempts to get a low rank sep-arated representation of the solution for SSP in a systematic enrichment manner.No iteration is performed at each enrichment step.In order to satisfy the inf–sup condition in the mixed formulation,we enrich the separated terms for the primal system variable at each enrichment step.For the SSP problems by regularization or penalty,we propose a more efficient variable-separation?VS?method,i.e.,the variable-separation by penalty method.This can avoid further enrichment of the separated terms in the original mixed formulation.Finally,we propose two reduced mixed Generalized multiscale finite element basis methods?RmGMsFEM?for the complicated models in heterogeneous ran-dom porous media.A typical application for the elliptic PDEs is the flow in heterogeneous random porous media.Mixed generalized multiscale finite element method?GMsFEM?is one of the accurate and efficient approaches to solve the flow problem in a coarse grid and obtain the velocity with local mass conservation.When the inputs of the PDEs are parameterized by the random variables,the GMsFE basis functions usually depend on the random parameters.This leads to a large number degree of freedoms for the mixed GMsFEM and substantially im-pacts on the computation efficiency.In order to overcome the difficulty,we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space.To this end,a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space.Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies:basis-oriented cross-validation and proper orthogonal decomposition.