Research On Numerical Methods Of Stochastic Multiscale Models

This thesis mainly studies three different dimensionality reduction methods for stochastic multiscale models.Under the framework of constrained energy minimization generalized multiscale finite element method(CEM-GMs FEM),combined with the uncertainty quantification method,more efficient and reasonable stochastic multiscale algorithms are proposed and the corresponding error estimations are given,so as to ensure a certain calculation accuracy.It can greatly reduce the amount of computation and storage requirements,and improve the computational efficiency.First,we study linear stochastic parabolic partial differential equations with multiscale diffusion coefficients whose source terms are driven by different time noises.The infinite dimensional additive noise is approximated as a finite dimensional noise in the Fourier form.Combined with CEM-GMs FEM,a stochastic multi-scale algorithm is designed and the convergence analysis and error estimation of the algorithm are given.In the semidiscrete formulation,we obtain that the convergence rate is related to the coarse grid size,the decay of eigenvalues for local spectral problems,and the number of truncations for random noise.Further,we use the backward Euler technique to discretize this equation in the time direction,and obtain the error estimate under the full discretization.Different examples are given to verify our theoretical analysis,and the numerical results demonstrate the rationality and computational efficiency of the algorithm.Next,we discuss the stochastic ellipse problem driven by high dimensional space noise and the stochastic parabolic problem driven by space time noise.Using CEMGMs FEM,we construct a novel multiscale spectral representation of high-dimensional noise,The effectiveness of multiscale spectral representation of noise is illustrated by comparing it with the noise represented in the ordinary Fourier form.Corresponding convergence analysis and error estimation are derived based on the form of mild solutions and variational form,and the influence of this special noise on multi-scale computational accuracy is demonstrated.Through the convergence analysis,it is concluded that the algorithm not only retains the advantages of CEM-GMs FEM,which is linearly dependent on coarse grids and independent of high contrast coefficient,but also can effectively describe the influence of these special noises on the solutions.The numerical results not only validate the theoretical results,but also reveal the efficient computational performance of the algorithm,which is an effective way to deal with position-dependent or position-time-coupling-dependent noises in complex multiscale stochastic physical systems.Finally,for the thermal conduction behavior of heterogeneous solids with uncertain material parameters,we innovatively propose an multi-modes based constrained energy minimization generalized multiscale finite element method(MCEM-GMs FEM).The original stochastic multiscale model with high-dimensional uncertain material parameters is transformed into a series of recursive multiscale models with the same deterministic multiscale material parameters by the multimodes expansion method.In particular,all multiscale models share the same coefficient matrices after multimodes expansion,and LU decomposition is designed to effectively reduce the complexity of repeated calculation of discrete multiscale systems.Furthermore,the convergence analysis is established and the optimal error estimates are derived.The theoretical results are verified in a numerical example by considering several typical random perturbations to the mean value of the uncertain thermal conductivity.Numerical results show that the multi-modes multiscale method is a robust integrated method with excellent performance,which can efficiently and accurately estimate the heat transfer effect of heterogeneous solids with high-dimensional uncertain material parameters,and can significantly reduce the computation time.