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

Ping Liu¹ ; C. I. Siettos² ; C. W. Gear³ ; I. G. Kevrekidis^3, 4

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

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 3, pp. 71-90.

Résumé

We study the coarse-grained, reduced dynamics of an agent-based market model due to Omurtag and Sirovich []. 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.

Détail

DOI : 10.1051/mmnp/201510307

Bibliographie

[1] E. Blanchart, N. Marilleau, J.L. Chotte, A. Drogoul, E. Perrier, C. Cambier European Journal of Soil Science 2009 13 21

[2] J. Buhl, D.J.T. Sumpter, I.D. Couzin, J.J. Hale, E. Despland, E.R. Miller, S.J. Simpson Science 2006 1402 1406

[3] F. Castiglione, F. Pappalardo, M. Bernaschi, S. Motta Bioinformatics 2007 3350 3355

[4] R.R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, S. Zucker Proceedings of the National Academy of Sciences of the United States of America 2005 7432 7437

[5] R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, B. Nadler, F. Warner, S.W. Zucker Proceedings of the National Academy of Sciences of the United States of America 2005 7426 7431

[6] A. Dhooge, W. Govaerts, Y.A. Kuznetsov ACM Trans. Math. Softw. 2003 141 164

[7] E.J. Doedel, R.C. Paffenroth, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B.E. Oldeman, B. Sandstede, X. Wang. AUTO 2000: Continuation and bifurcation software for ordinary differential equaitons (with HomCont). Technical Report. California Institute of Technology, Pasadena, CA, 2001.

[8] D.T. Gillespie. Markov Process. Academic Press, San Diego, 1992.

[9] L. Hamill, N. Gilbert The Journal of Artificial Societies and Social Simulation 2009 3

[10] F.L. Hellweger, V. Bucci Ecological Modelling 2009 8 22

[11] E. Ilie-Zudor, L. Monostori Assembly Automation 2009 137 153

[12] M.A. Janssen, L.N. Alessa, M. Barton, S. Bergin, A. Lee The Journal of Artificial Societies and Social Simulation 2008 6

[13] I.G. Kevrekidis, C.W. Gear, G. Hummer AIChE Journal 2004 1346 1355

[14] I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, C. Theodoropoulos Comm. Math. Sciences 2003 715 762

[15] I.G. Kevrekidis, G. Samaey Annual Review of Physical Chemistry 2009 321 344

[16] B. Nadler, S. Lafon, R.R. Coifman, I.G. Kevrekidis Applied and Computational Harmonic Analysis 2006 113 127

[17] I. Nishizaki, H. Katagiri, T. Oyama Computational Economics 2009 37 65

[18] A. Omurtag, L. Sirovich Journal of Economic Behavior & Organization 2006 562 576

[19] A. Pan, S.Y.S. Leung, K.L. Moon, K.W. Yeung Expert Systems with Applications 2009 8571 8581

[20] R. Pinnau. Model Reduction via Proper Orthogonal Decomposition. In Model Order Reduction: Theory, Research Aspects and Applications. W. A. Schilders, H. van der Vorst, J. Rommes, Eds. vol. 13. Springer, Berlin Heidelberg, (2008), 95-109.

[21] H. Risken. The Fokker-Planck Equation. Methods of Solution and Applications. Second edition. Springer, Berlin, 1989.

[22] O. Runborg, C. Theodoropoulos, I.G. Kevrekidis Nonlinearity 2002 491 511

[23] E. Samanidou, E. Zschischang, D. Stauffer, T. Lux Reports on Progress in Physics 2007 409 450

[24] T. Shimokawa, K. Suzuki, T. Misawa Physica A 2007 207 225

[25] C.I. Siettos, C.W. Gear, I.G. Kevrekidis Europhys. Lett. 2012 48007

[26] J.B. Tenenbaum, V. De Silva, J.C. Langford Science 2000 2319

[27] B.C. Thorne, A.M. Bailey, S.M. Peirce Briefings in Bioinformatics 2007 245 257

[28] D. Tykhonov, C. Jonker, S. Meijer, T. Verwaart The Journal of Artificial Societies and Social Simulation 2008 1

[29] F.H. Westerhoff Jahrbucher Fur Nationalokonomie Und Statistik 2008 195 227

[30] L. Zhang, Z.H. Wang, J.A. Sagotsky, T.S. Deisboeck Journal of Mathematical Biology 2009 545 559