Equation-Free Analysis for Agent-Based Computation

Abstract

Due to the advances in computing technology, agent-based modeling (ABM) has become a powerful tool in addressing a wide range of problems. However, there is a challenge which modelers often encounter: the effective nonlinear and stochastic nature of individual dynamics and the inherent complexity of microscopic descriptions; closures that allow us to write macroscopic evolution equations for the coarse-grained dynamics are usually not available.

As an attempt to overcome this difficulty and to enable system-level analysis of agent-based simulations, the Equation-Free (EF) approach is explored in this Thesis in studying two different agent-based models. The first agent-based model describes the dynamic behavior of many interacting investors in a financial market in the presence of mimesis. Three aspects of the EF framework are successfully applied to this model: (1) in the coarse bifurcation analysis, using appropriately initialized short runs of the microscopic agent-based simulations, bifurcation diagrams of the identified

coarse variables are constructed, and the stability of its multiple solution branches are determined; (2) in the coarse rare event analysis, an effective Fokker-Planck (FP) equation is constructed on a coarse-grained one dimensional reaction coordinate. The mean escape time of the associated rare event computed using this effective FP shows good agreement with the results from direct agent-based simulations, but requires only 3.2% of the computational time; (3) utilizing the smoothness of coarse variables, a patch-dynamics scheme is successfully designed which allows expensive agent-based simulations to be performed in small "patches" (2%) of the full spatio-temporal domain, while giving comparable system-level solutions.

The second agent-based model describes the dynamic behavior of a group of swarming animals. Using a recently developed data-mining technique - Diffusion Maps (DMAP) - interesting coarse level features about the swarming dynamics were successfully captured. The first two dominant DMAP coarse variables characterize the "up-down" and "left-right" directions of collective group motion respectively. Based on these two DMAP coarse variables, a reduced stochastic differential equation (SDE) model is successfully constructed using the EF framework. Using the reduced SDE model, the associated coarse rare events are efficiently studied, circumventing expensive long-term agent-based simulations.