Final Activity Report Summary - HI-CG (Mathematical strategies towards hierarchical coarse-grainings of multiscale systems) 1. Error quantification and adaptive coarse-graining for lattice systems.The first aspect of our work builds on our earlier publications: [M. A. Katsoulakis, P. Plechac, A. Sopasakis, SIAM Num. Anal. 2006], and [M. A. Katsoulakis,J. Trashorras J. Stat. Phys. 2006]Here we investigated the approximation properties of the coarse-graining procedure and provided both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allowed for error control of quantities that may also depend on the path, in both transient and long-time simulations. We demonstrated that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate was shown to hold in the weak convergence sense and in relative entropy. We employed the derived analytical results to guide CGMC algorithms and we demonstrated a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability. We also introduced a critical mathematical tool for our analysis that allowed us to reconstruct microscopic paths from coarse grained simulations with controlled error.a. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems b. Mathematical strategies in the coarse-graining of extensive systems: error quantification and adaptivityc. Multi-body interactions in coarse-graining schemes for extended systemsd. Numerical and Statistical Methods for the Coarse-Graining of Many-Particle Stochastic Systemse. Coarse-Grained Langevin approximations and spatiotemporal acceleration for Kinetic Monte Carlosimulations of diffusion of interacting particles2. We study a class of model prototype hybrid systems comprised of a microscopicArrhenius surface process modelling adsorption/desorption and/or surface diffusion of particles coupled to an ordinary differential equation displaying bifurcations triggered by the microscopic process. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic sciences.3. Cluster evolution in surface processes Surface processes, such as catalysis, deposition and epitaxial growth, typically involve transport and chemistry of precursors in a gas phase; unconsumed reactants and radicals adsorb onto the surface of a substrate where numerous processes that take place simultaneously, for instance surface diffusion, reactions and desorption. In the JDE paper we study the effect of multiple micro-mechanisms on macroscopic cluster interface morphology and dynamics. We also study the asymptotics for a generalised Cahn-Hilliard equation with forcing terms.4. Extended field theories in polymers. In this work we construct a model reduction procedure for the study of equilibrium propertiesof statistical mechanical models, focusing on complex polymeric materials models.5. A Comparison Principle for Hamilton-Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions.6. Numerical approximation of stochastic interfacesOther activitiesPostdoctoral fellow funding and mentoringDr. Evaggelia Kalligianaki is a post-doctoral fellow funded by the project. She is a recent PhD graduate of the Department of Applied Mathematics of the University of Crete. After the end of the IRG grant that supported her, Dr. Kalligianaki took a position as researcher at the prestigious Oak Ridge National Laboratory in the US.A starting September 1, 2009.b. WorkshopsOn June 25-27, 2007 we organised a small scale workshop on "Mathematical and Computational Methods for Accelerated Molecular, Stochastic and Hybrid Simulation".