Multiscale Partial Differential Equations Research Articles

Multiscale empirical interpolation for solving nonlinear PDEs

In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to represent the solution on a coarse grid with multiscale basis functions computed offline. Computing the GMsFEM solution involves calculating the system residuals and Jacobians on the fine grid. We use empirical interpolation concepts to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one. The empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. The proposed multiscale empirical interpolation techniques: (1) divide computing the nonlinear function into coarse regions; (2) evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution; and (3) introduce multiscale proper-orthogonal-decomposition techniques to find appropriate interpolation vectors. We demonstrate the effectiveness of the proposed methods on several nonlinear multiscale PDEs that are solved with Newton's methods and fully-implicit time marching schemes. Our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale PDEs on a coarse grid with bounded error and significant computational cost reduction.

A probabilistic graphical model approach to stochastic multiscale partial differential equations

We develop a probabilistic graphical model based methodology to efficiently perform uncertainty quantification in the presence of both stochastic input and multiple scales. Both the stochastic input and model responses are treated as random variables in this framework. Their relationships are modeled by graphical models which give explicit factorization of a high-dimensional joint probability distribution. The hyperparameters in the probabilistic model are learned using sequential Monte Carlo (SMC) method, which is superior to standard Markov chain Monte Carlo (MCMC) methods for multi-modal distributions. Finally, we make predictions from the probabilistic graphical model using the belief propagation algorithm. Numerical examples are presented to show the accuracy and efficiency of the predictive capability of the developed graphical model.

The Multiscale Finite Element Method with Nonconforming Elements for Elliptic Homogenization Problems

The multiscale finite element method was developed by Hou and Wu [J. Comput. Phys., 134 (1997), pp. 169–189] to capture the effect of microscales on macroscales for multiscale problems through modification of finite element basis functions. For second-order multiscale partial differential equations, continuous (conforming) finite elements have been considered so far. Efendiev, Hou, and Wu [SIAM J. Numer. Anal., 37 (2000), pp. 888–910] considered a nonconforming multiscale finite element method where nonconformity comes from an oversampling technique for reducing resonance errors. In this paper we study the multiscale finite element method in the context of nonconforming finite elements for the first time. When the oversampling technique is used, a double nonconformity arises: one from this technique and the other from nonconforming elements. An equivalent formulation recently introduced by Chen [Numer. Methods Partial Differential Equations, 22 (2006), pp. 317–360] (also see [Y. R. Efendiev, T. Hou, and V. Ginting, Commun. Math. Sci., 2 (2004), pp. 553–589]) for the multiscale finite element method, which utilizes standard basis functions of finite element spaces but modifies the bilinear (quadratic) form in the finite element formulation of the underlying multiscale problems, is employed in the present study. Nonlinear multiscale and random homogenization problems are also studied, and numerical experiments are presented.

Numerical solution of differential equations using Haar wavelets

Haar wavelet techniques for the solution of ODE and PDE is discussed. Based on the Chen–Hsiao method [C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.—Control Theory Appl. 144 (1997) 87–94; C.F. Chen, C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146 (1997) 213–219] a new approach—the segmentation method—is developed. Five test problems are solved. The results are compared with the result obtained by the Chen–Hsiao method and with the method of piecewise constant approximation [C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulat. 57 (2001) 347–353; S. Goedecker, O. Ivanov, Solution of multiscale partial differential equations using wavelets, Comput. Phys. 12 (1998) 548–555].

Solution of Multiscale Partial Differential Equations Using Wavelets

Wavelets are a powerful new mathematical tool which offers the possibility to treat in a natural way quantities characterized by several length scales. In this article we will show how wavelets can be used to solve partial differential equations which exhibit widely varying length scales and which are therefore hardly accessible by other numerical methods. As a benchmark calculation we solve Poisson's equation for a 3-dimensional Uranium dimer. The length scales of the charge distribution vary by 4 orders of magnitude in this case. Using lifted interpolating wavelets the number of iterations is independent of the maximal resolution and the computational effort therefore scales strictly linearly with respect to the size of the system.