Problems in diverse areas of scientific, technological, and societal relevance, ranging from the development of novel materials to understanding fundamental biological mechanisms to weather and climate prediction, are intrinsically multi-scale. In essence this means that information at very small spatial and temporal scales, for instance properties and motions of atoms or molecules, can profoundly impact intermediate- and large-scale behavior, such as the permeability of a membrane or the viscosity of a fluid. From a modeling and computational perspective, small spatiotemporal scales can be described in detail with microscopic models such as Molecular Dynamics or Monte Carlo methods that typically account for atomistic and/or molecular information. On the other hand, understanding large-scale, macroscopic properties of materials requires simulations with microscopic systems at prohibitively large spatial (e.g. 10^23 atoms), as well as temporal scales. One such example arises in the design of novel materials with pre-specified properties, where numerical simulations--when feasible--could be used as a flexible and inexpensive predictive tool. An important class of computational tools that has been developed in recent years, precisely to bridge such scales gaps by speeding up microscopic simulation methods, is the method of coarse-graining. The idea of this approach is to reduce the molecular system''s complexity by lumping together degrees of freedom into coarse-grained variables, thus yielding an accelerated simulation methodology. Such coarse-grained models have been developed for the study and simulation of micro-reactors (e.g., portable energy sources), polymers, proteins, and bio-fluids (e.g., red-cell flow in small blood vessels), among others. Existing approaches can give unprecedented speed-ups to molecular simulations and can work well in certain parameter regimes, such as high temperatures. On the other hand, they can also give wrong predictions on important features such as diffusion, crystallization, and phase transitions. Along these lines, a relevant mathematical and statistical goal to numerous applications, such as the ones mentioned earlier, is to develop systematic diagnostics for determining when coarse-graining methods can give reliable predictions, and how they can be further enhanced. Indeed, in our proposed work we intend to carry out two related main tasks: (a) understand the validity regimes of existing coarse-graining methods by developing a mathematical and statistical error quantification analysis; and (b) develop improved algorithms capable of operating in much wider parameter regimes, with the capacity to automatically adjust once substantial deviations are detected during simulation. In our proposed work, we plan to develop novel multi-scale mathematical and computational methods, bringing together diverse techniques from probability theory, statistical mechanics, information theory, statistics, and finite elements. A critical component of the proposed work relies on the synergy between applied mathematics and statistics methods, operating in a complementary fashion, that can provide a mathematically systematic framework for developing flexible and reliable coarse-graining algorithms for molecular simulations.
