Equation-free computation in multiscale/complex systems modeling: Enabling microscopic simulators to perform system-level tasks

Yannis G. Kevrekidis, Chemical Engineering, Program in Applied and Computational Mathematics, and Mathematics, Princeton University

Traditional modeling approaches start by deriving macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in PNAS (2000) when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form.

Macroscopic numerical analysis algorithms thus become protocols for the design and processing of computational experiments using the detailed, microscopic system description. Appropriately designed and initialized short bursts of microscopic/stochastic simulation are used to estimate on the fly (?on demand?) the quantities required for macroscopic analysis. Systems theory tools (variance reduction, estimation, filtering) complete the loop in this system identification based, "closure on demand" computational enabling technology

Our computational superstructures can be thought of as mathematically inspired alternative computational ensembles, aimed at efficiently extracting macroscopic information with the minimum of computational experimentation at the detailed microscopic / atomistic / agent-based level. It is interesting that, under some circumstances, these protocols for computational experimentation we will discuss and illustrate, can also become efficient protocols for alternative laboratory experimentation too.