Initializing at will: Equation-free modeling and experimentation

Yannis G. Kevrekidis,
Chemical Engineering, PACM and Mathematics, Princeton University

In current modeling, the best available descriptions of a system come at a fine level (atomistic, stochastic, microscopic, individual-based) while the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level. Traditional modeling approaches start by first deriving macroscopic evolution equations from the microscopic models, and then bringing our arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions.
Over the last few years, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the "equation-free" approach, since it circumvents the step of obtaining accurate macroscopic descriptions.
Illustrative applications include coarse kinetic Monte-Carlo computations of bacterical chemotaxis and particle hydrodynamics as well as coarse Molecular Dynamics of alanine dipeptide folding. Time permitting, we will also discuss coarse equilibrium Monte-Carlo computations (micelle formation) as well as coarse dynamic renormalization techniques.