Mathematical strategies in the coarse-graining of extensive systems: Error quantification and adaptivity
Publication Date
2008
Journal or Book Title
JOURNAL OF NON-NEWTONIAN FLUID MECHANICS
Abstract
In this paper we continue our study of coarse-graining schemes for stochastic many-body microscopic models started in Katsoulakis et al. [M. Katsoulakis, A. Majda, D. Vlachos, Coarse-grained stochastic processes for microscopic lattice systems, Proc. Natl. Acad. Sci. 100 (2003) 782–782, M.A. Katsoulakis, L. Rey-Bellet, P. Plecháč, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press], focusing on equilibrium stochastic lattice systems. Using cluster expansion techniques we expand the exact coarse-grained Hamiltonian around a first approximation and derive higher accuracy schemes by including more terms in the expansion. The accuracy of the coarse-graining schemes is measured in terms of information loss, i.e., relative entropy, between the exact and approximate coarse-grained Gibbs measures. We test the effectiveness of our schemes in systems with competing short- and long-range interactions, using an analytically solvable model as a computational benchmark. Furthermore, the cluster expansion in Katsoulakis et al. [M.A. Katsoulakis, L. Rey-Bellet, P. Plecháč, D. Tsagkarogiannis, Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems, M2AN Math. Model. Numer. Anal., in press] yields sharp a posteriori error estimates for the coarse-grained approximations that can be computed on-the-fly during the simulation. Based on these estimates we develop a numerical strategy to assess the quality of the coarse-graining and suitably refine or coarsen the simulations. We demonstrate the use of this diagnostic tool in the numerical calculation of phase diagrams.
Pages
101-112
Volume
152
Issue
1-3
Recommended Citation
Katsoulakis, MA; Plechac, P; Rey-Bellet, L; and Tsagkarogiannis, DK, "Mathematical strategies in the coarse-graining of extensive systems: Error quantification and adaptivity" (2008). JOURNAL OF NON-NEWTONIAN FLUID MECHANICS. 440.
Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/440
Comments
The published version is located at http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGV-4NSMMP9-1&_user=1516330&_coverDate=06%2F30%2F2008&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1672114267&_rerunOrigin=google&_acct=C000053443&_version=1&_urlVersion=0&_userid=1516330&md5=a7ac75d53b0f02be426db06fb1f18e88&searchtype=a