TY - JOUR

T1 - Equation-free model reduction in agent-based computations

T2 - Coarse-grained bifurcation and variable-free rare event analysis

AU - Liu, Ping

AU - Siettos, C. I.

AU - Gear, C. W.

AU - Kevrekidis, I. G.

N1 - Publisher Copyright:
© 2015 EDP Sciences.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Agent based modeling

KW - Coarse-graining

KW - Equation-free computation

KW - Rare events

UR - http://www.scopus.com/inward/record.url?scp=84937436650&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937436650&partnerID=8YFLogxK

U2 - 10.1051/mmnp/201510307

DO - 10.1051/mmnp/201510307

M3 - Article

AN - SCOPUS:84937436650

SN - 0973-5348

VL - 10

SP - 71

EP - 90

JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

IS - 3

ER -