Commun. Comput. Phys., Notice: Undefined index: year in /var/www/html/issue/abstract/readabs.php on line 20 Notice: Undefined index: ppage in /var/www/html/issue/abstract/readabs.php on line 21 Notice: Undefined index: issue in /var/www/html/issue/abstract/readabs.php on line 23 Volume 4. 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 Abstract 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 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Multiscale modeling, sparse grids. *Corresponding author. Email: cgarcia@math.ucsb.edu (C. J. Garcia-Cervera), weiqing@cims.nyu.edu (W. Ren), jianfeng@math.Princeton.EDU (J. Lu), weinan@Princeton.EDU} (W. E)