<< Previous                    Main Menu                            Next >>

Markos A. Katsoulakis, Petr Plechac, Luc Rey-Bellet, Dimitrios K. Tsagkarogiannis

Mathematical strategies in the coarse-graining of extensive systems: error quantification and adaptivity

In this paper we continue our study of coarse-graining schemes for stochastic many-body microscopic models started in \cite{KMVPNAS, KPRT}, 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 \cite{KPRT} 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.

Bibliographical note: J. Non-Newtonian Fluid Mech., in press, available on-line 21 May, 2007