Commun. Comput. Phys.,
Sequential Multiscale Modeling Using Sparse Representation
Carlos J. Garcia-Cervera 1*, Weiqing Ren 2, Jianfeng Lu 3, Weinan E 41 Mathematics Department, University of California, Santa Barbara, CA 93106, USA.
2 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
3 Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA.
4 Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA.
Received 11 February 2008; Accepted (in revised version) 26 March 2008
Available online 8 July 2008
The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relation which often involves many independent variables. The constitutive relation of a polymeric fluid is a function of six variables, even after making the simplifying assumption that stress depends only on the rate of strain. Precomputing such a function is usually considered too expensive. Consequently the value of sequential multiscale modeling is often limited to ``parameter passing''. Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables. This strategy dramatically increases the efficiency of sequential multiscale modeling, making it very competitive in many situations.AMS subject classifications: 65Z05, 35Q30, 35Q72
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Key words: Multiscale modeling, sparse grids.
Email: email@example.com (C. J. Garcia-Cervera), firstname.lastname@example.org (W. Ren), jianfeng@math.Princeton.EDU (J. Lu), weinan@Princeton.EDU} (W. E)