Equation-free, coarse-grained computational optimization using timesteppers

System level optimization computations for engineering problems are typically based on continuum level, macroscopic system descriptions, obtained using accurate closures. In many cases, however, including micro/nanoscopic systems, the best available description is a fine scale (atomistic, stochastic or agent-based) model for which accurate, coarse-grained, system level descriptions are not known. The recently introduced equation-free approach [Theodoropoulos, K., Qian, Y.-H., Kevrekidis, I.G., 2000. "Coarse" stability and bifurcation analysis using timesteppers: a reaction diffusion example. Proceedings of the National Academy of Sciences 97, 9840-9843; Gear, C.W., Kevrekidis, I.G., Theodoropoulos, C., 2002. ‘Coarse’ integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Computers and Chemical Engineering 26, 941-963; Kevrekidis, I.G., Gear, C.W., Hummer, G., 2004. Equation-free: the computer-assisted analysis of complex, multiscale systems. A.I.Ch.E. Journal 50, 1346-1354; Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, K., 2003. Equation-free multiscale computation: enabling microscopic simulators to perform system-level tasks. Communications in Mathematical Sciences 1, 715-762] provides a computational bridge between the underlying microscopic process model and system level numerical computations. In this paper, we employ the equation-free approach to perform system level optimization by acting directly on microscopic/stochastic models. The approach substitutes the evaluation of closed form macroscopic equations with the design and execution of appropriately initialized short bursts of fine scale simulation; processing the simulation results yields estimates of the quantities (residuals, actions of Jacobians and Hessians) required for continuum computations. We illustrate the combination of "coarse timesteppers" with standard (both local and global) optimization techniques. The efficiency of alternative optimization formulations is compared; we see that it can be enhanced by exploiting a separation of time-scales in the system dynamics. The approach constitutes a computational "wrapper" around microscopic/stochastic simulators; yet it can also be wrapped around legacy continuum dynamic simulators.