TY - JOUR

T1 - EQUATION-FREE, COARSE-GRAINED MULTISCALE COMPUTATION

T2 - ENABLING MICROSCOPIC SIMULATORS TO PERFORM SYSTEM-LEVEL ANALYSIS*

AU - Kevrekidis, Ioannis G.

AU - Gear, C. William

AU - Hyman, James M.

AU - Kevrekidis, Panagiotis G.

AU - Runborg, Olof

AU - Theodoropoulos, Constantinos

N1 - Funding Information:
Acknowledgements. The work reported in this paper spans several years and involves interactions with several people, whom we would like to acknowledge. Original discussions with Herb Keller about RPM (at the IMA in Minneapolis) and with Jim Evans at Iowa State about his hybrid MC simulators of surface reactions played an important role. Discussions over the years with Profs. J. Keller, K. Lust, D. Maroudas, D. McLaughlin, A. Majda, L. Ju, S. Yip, R. Armstrong, A. Panagiotopou-los, M. Graham, R. Kapral, H.-C. Oettinger, V. Yakhot, C. Foias, E. Titi, P. Constantin, M. Aizenman, J. Burns, F. Alexander, N. Hadjiconstantinou, A. Makeev, C. Siettos, C. Pantelides, C. Jacobsen, and A. Armaou have affected this work. In the course of the last three years we have discussed this research direction with our colleagues, Professors Engquist and E at PACM in Princeton. In the companion paper [119] they present an alternative formulation for conservation laws which, in the time-dependent case, corresponds to a generalized Godunov scheme. The role of AFOSR (Dynamics and Control, Drs. Jacobs and King), to whom this work was proposed in 1999, and which they supported through the years, has been crucial. Partial support by NSF through a KDI and an ITR grant, by DARPA, by a Humboldt Forschungspreis to I.G.K, and Los Alamos National Lab (through LDRD-DR-2001501) are also gratefully acknowledged.
Publisher Copyright:
© 2003 International Press

PY - 2003

Y1 - 2003

N2 - We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in [1], when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,“coarse” bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, “closure-on-demand” computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We will briefly survey the application of these “numerical enabling technology” ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

AB - We present and discuss a framework for computer-aided multiscale analysis, which enables models at a fine (microscopic/stochastic) level of description to perform modeling tasks at a coarse (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Traditional modeling approaches first involve the derivation of macroscopic evolution equations (balances closed through constitutive relations). An arsenal of analytical and numerical techniques for the efficient solution of such evolution equations (usually Partial Differential Equations, PDEs) is then brought to bear on the problem. Our equation-free (EF) approach, introduced in [1], when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,“coarse” bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a system identification based, “closure-on-demand” computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We will briefly survey the application of these “numerical enabling technology” ideas through examples including the computation of coarsely self-similar solutions, and discuss various features, limitations and potential extensions of the approach.

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U2 - 10.4310/CMS.2003.v1.n4.a5

DO - 10.4310/CMS.2003.v1.n4.a5

M3 - Article

AN - SCOPUS:85128858890

SN - 1539-6746

VL - 1

SP - 715

EP - 762

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 4

ER -