Chemical Engineering Science 61 (2006) 779 – 793 www.elsevier.com/locate/ces Equation-free, coarse-grained computational optimization using timesteppers Aditya Bindal a , Marianthi G. Ierapetritou a , Suhrid Balakrishnan a , Antonios Armaou b , Alexei G. Makeev c , Ioannis G. Kevrekidis d , ∗ a Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ 08854, USA b Chemical Engineering Department, The Pennsylvania State University, University Park, PA 16802, USA c Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119899, Russia d Departments of Chemical Engineering, PACM and Mathematics, Princeton University, Princeton, NJ 08544, USA Received 5 January 2005; received in revised form 16 June 2005; accepted 28 June 2005 Available online 19 September 2005 Abstract System level optimization computations for engineering problems are typically based on continuum level, macroscopic system descrip- tions, 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 optimiza- tion 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 combi- nation 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.  2005 Elsevier Ltd. All rights reserved. Keywords: Optimization; Numerical analysis; Equation-free; Timestepper 1. Introduction The dynamics of many physicochemical systems occur across different length scales, often classified as micro- scopic, mesoscopic and macroscopic. Traditionally, when ∗ Corresponding author. Tel.: +1 609 258 2818; fax: +1 609 258 0211. E-mail address: yannis@Princeton.edu (I.G. Kevrekidis). 0009-2509/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.06.034 analyzing and designing engineering systems we are inter- ested at the coarse-grained, macroscopic, systems level be- havior and performance; the models that are used for these tasks are evolution equations (e.g., mass, momentum and energy balances) closed through constitutive equations (e.g., Newton’s law of viscosity, chemical kinetics). At the atom- istic, fine scale, the dynamics involve the evolution of inter- acting entities (molecules); yet the macroscopic equations