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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2015014280
pages 1-23

SPARSE GENERALIZED MULTISCALE FINITE ELEMENT METHODS AND THEIR APPLICATIONS

Eric T. Chung
Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong SAR, China
Yalchin Efendiev
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77840, USA; Multiscale Model Reduction Laboratory, North-Eastern Federal University, Yakutsk, Russia, 677980
Wing Tat Leung
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
Guanglian Li
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas, USA; Institute for Numerical Simulation, The University of Bonn, Wegelerstrasse 6,53115 Bonn, Germany

ABSTRACT

In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1 minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1 minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.


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