Equation-free Model Reduction in Agent-based Computations: Coarse-grained Bifurcation and Variable-free Rare Event Analysis

¹ Department of Molecular, Cellular and Developmental Biology, Yale University, West Haven, CT 06516, USA
² School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, Greece
³ Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA
⁴ Program in Applied and Computational Mathematics (PACM), and Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA

^⋆ Corresponding author. E-mail: yannis@princeton.edu

Abstract

We study the coarse-grained, reduced dynamics of an agent-based market model due to Omurtag and Sirovich [18]. We first describe the large agent number, deterministic limit of the system dynamics by performing numerical bifurcation calculations on a continuum approximation of their model. By exploring a broad parameter space, we observe several interesting phenomena including turning points leading to unstable stationary agent density distributions as well as a type of “termination point.” Close to these deterministic turning points we expect the stochastic underlying model to exhibit rare event transitions. We then proceed to discuss a coarse-grained approach to the quantitative study of these rare events. The basic assumption is that the dynamics of the system can be decomposed into fast (noise) and slow (single reaction coordinate) dynamics, so that the system can be described by an effective, coarse-grained Fokker-Planck(FP) equation. An explicit form of this effective FP equation is not available; in our computations we bypass the lack of a closed form equation by numerically estimating its components - the drift and diffusion coefficients - from ensembles of short bursts of microscopic simulations with judiciously chosen initial conditions. The reaction coordinate is first constructed based on our understanding of the continuum model close to the turning points, and it gives results reasonably close to those from brute-force direct simulations. When no guidelines for the selection of a good reaction coordinate are available, data-mining tools, in particular Diffusion Maps, can be used to determine a suitable reaction coordinate. In the third part of this work we demonstrate this “variable-free” approach by constructing a reaction coordinate simply based on the data from the simulation itself. This Diffusion Map based, empirical coordinate gives results consistent with the direct simulation.