IOANNIS G. KEVREKIDIS

(WITH C.W. GEAR AND G. HUMMER)

*Department of Chemical Engineering, Program in Applied and Computational Mathematics, and Department of Mathematics*

*Princeton University*

*Princeton, New Jersey*

*In current modeling, the best available descriptions of a system are often given at a fine level (atomistic, stochastic, microscopic, individual-based) even though the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level***.** *Traditional modeling approaches first derive macroscopic evolution equations from the microscopic models and then bring an arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions. Over the last few years, and with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the “equation-free” approach because it circumvents the step of obtaining accurate macroscopic descriptions. We argue that the basis of this approach is the design of (computational) experiments. Traditional continuum numerical algorithms can be viewed as protocols for experimental design (where “experiment” means a computational experiment set up and performed with a model at a different level of description). Ultimately, what makes the equation-free approach possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized*

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Tenth Annual Symposium on Frontiers of Engineering
Equation-Free Modeling for Complex Systems
IOANNIS G. KEVREKIDIS
(WITH C.W. GEAR AND G. HUMMER)
Department of Chemical Engineering, Program in Applied and Computational Mathematics, and Department of Mathematics
Princeton University
Princeton, New Jersey
ABSTRACT
In current modeling, the best available descriptions of a system are often given at a fine level (atomistic, stochastic, microscopic, individual-based) even though the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level. Traditional modeling approaches first derive macroscopic evolution equations from the microscopic models and then bring an arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions. Over the last few years, and with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the “equation-free” approach because it circumvents the step of obtaining accurate macroscopic descriptions. We argue that the basis of this approach is the design of (computational) experiments. Traditional continuum numerical algorithms can be viewed as protocols for experimental design (where “experiment” means a computational experiment set up and performed with a model at a different level of description). Ultimately, what makes the equation-free approach possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized

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computational experimentation—through matrix-free numerical analysis and systems theory tools like variance reduction and estimation—bridge microscopic simulation with macroscopic modeling. Remarkably, if there is enough control authority to initialize laboratory experiments “at will,” this computational enabling technology can become a set of experimental protocols for the equation-free exploration of complex system dynamics.
A persistent feature of many complex systems is the emergence of macroscopic, coherent behavior from the interactions of microscopic “agents”—molecules, cells, individuals in a population—among themselves and with their environment. The implication is that macroscopic rules, a description of the system at a coarse-grained, high level, can somehow be deduced from microscopic rules, a description at a much finer level. For laminar Newtonian fluid mechanics, a successful coarse-grained description, the Navier-Stokes equations, was known on a phenomenological basis long before its approximate derivation from kinetic theory. Today we must frequently study systems for which the physics can be modeled at a microscopic, fine scale; but for whose macroscopic behavior explicit equations are practically impossible to derive. Hence, we look to the computer to explore the macroscopic behavior based on the microscopic description.
Macroscopic models of reaction and transport processes in our textbooks are in the form of conservation laws (e.g., species, mass, momentum, energy) closed through constitutive equations (e.g., reaction rates as a function of concentration, viscous stresses as functionals of velocity gradients). These models are written directly at the scale (alternatively, at the level of complexity) at which we are interested in modeling the system behavior. Because we observe the system at the level of concentrations or velocity fields, we sometimes forget that what really evolves during an experiment is distributions of colliding and reacting molecules.
We know from experience with particular classes of problems that it is possible to write predictive, deterministic laws for the behavior (predictive over relevant space/time scales that are useful in engineering practice) observed at the level of concentrations or velocity fields. Knowing the right level of observation at which we can be practically predictive, we attempt to write closed evolution equations for the system at this level. The closures may be based on experiment (e.g., through engineering correlations) or on mathematical modeling and approximation of what happens at more microscopic scales (e.g., the Chapman-Enskog expansion).
In many problems of current modeling practice, ranging from materials science to ecology and from engineering to computational chemistry, the physics are known at the microscopic/individual level, but the closures required to translate them to a high-level, coarse-grained, macroscopic description are not available. Sometimes we do not even know at what level of observation one can

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be practically predictive. Severe computational limitations arise in trying to bridge, through direct computer simulation, the enormous gap between the scale of the available description and the macroscopic, “system” scale at which the questions of interest are asked and the practical answers are required (see, e.g. Maroudas, 2000; Lu and Kaxiras, 2004). These computational limitations are a major stumbling block in current complex system modeling.
We will describe a computational approach for dealing with any complex, multiscale system whose collective, coarse-grained behavior is simple when we know in principle how to model the system at a very fine scale (e.g., through molecular dynamics). We assume that we do not know how to write good simple model equations at the right coarse-grained, macroscopic scale for the collective, coarse-grained behavior. We will argue that, in many cases, the derivation of macroscopic equations can be circumvented—that by using short bursts of appropriately initialized microscopic simulation, one can effectively solve the macroscopic equations without ever writing them down. Thus, a direct bridge can be built between microscopic simulation (e.g., kinetic Monte Carlo [kMC], agent-based modeling) and traditional continuum numerical analysis. It is possible to enable microscopic simulators to directly perform macroscopic, system-level tasks.
The main idea is to consider the microscopic, fine-scale simulator as a (computational) experiment that one can set up, initialize, and run at will. Based on the results of appropriately designed, initialized, and executed brief computational experiments, we can estimate the same information that a macroscopic model as we would evaluate from explicit formulas.
The heart of the approach can be conveyed through a simple example. Consider a single, autonomous ordinary differential equation,
Think of it as a model for the dynamics of a reactant concentration in a stirred reactor. Equations like this embody “practical determinism”: given a finite amount of information—the state at the present time, c(t=0)—we can predict the state at a future time. Consider how this is done on the computer using, for illustration, the simplest numerical integration scheme, forward Euler:
Starting with the initial condition, c0, we go to the equation and evaluate f(c0), the time derivative, or slope of the trajectory c(t). We then use this value to make a prediction of the state of the system at the next time step, c1. We then repeat the process: go to the equation with c1 to evaluate f(c1) and use the Euler scheme to predict c2 and so on.

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Forgetting for the moment accuracy and adaptive step-size selection, consider how the equation is used: given the state, we evaluate the time derivative; and then, using mathematics (in particular, Taylor series) and smoothness to create a local linear model of the process in time, we make a prediction of the state at the next time step. A numerical integration code will “ping” a subroutine with the current state as input and will obtain as output the time derivative at this state. The code will then process this value and use local Taylor series to make a prediction of the next state (the next value of c at which to call the subroutine evaluating the function f).
Three simple things are important to notice. First, the task at hand (numerical integration) does not need a closed formula for f(c)—it only needs f(c) evaluated at a particular sequence of values cn. Whether the subroutine evaluates f(c) from a single-line formula, uses a table lookup, or solves a large subsidiary problem makes no difference from the point of view of the integration code. Second, the sequence of values cn at which we need the time derivative evaluated is not known a priori. It is generated as the task progresses, from processing the results of previous function evaluations through the Euler formula. We know that protocols exist for designing experiments to accomplish tasks such as parameter estimation (Box et al., 1978). In the same spirit, we can think of the Euler method, and of explicit numerical integrators in general, as protocols for specifying where to perform function evaluations based on the task we want to accomplish (computation of a temporal trajectory). Last, the form of the protocol (the Euler method here) is based on mathematics, particularly on smoothness and Taylor series. The trajectory is locally approximated as a linear function of time; the coefficients of this function are obtained from the model using function evaluations.
Suppose now that we do not have the equation, but we have the experiment itself. We can fill up the stirred reactor with reactant at concentration c0, run it for some time, and record the time series of c(t). Using the results of a short run (over, say, one minute), we can now estimate the slope, dc/dt at t=0, and predict (using the Euler method) where the concentration will be in, say 10 minutes. Now, instead of waiting nine minutes for the reactor to get there, we stop the experiment and immediately start a new one. We reinitialize the reactor at the predicted concentration, run for one more minute, and use forward Euler to predict what the concentration will be 20 minutes down the line. We are substituting short, appropriately initialized experiments, and estimation based on the experimental results, for the function evaluations that the subroutine with the closed form f(c) would return. We are in effect doing forward Euler again, but the coefficients of the local linear model are obtained using experimentation “on demand” (Cybenko, 1996) rather than function evaluations of an a priori available model.
Now we complete the argument. Suppose that the inner layer is not a laboratory experiment but a computational experiment with a model at a different,

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much finer level of description—for the sake of the discussion, a lattice kMC model of the reaction. Instead of running the kMC model for long times and observing the evolution of the concentration, we can exploit the procedure described above, perform only short bursts of appropriately initialized microscopic simulation, and use the results to evolve the macroscopic behavior over longer time scales.
It is much easier to initialize a code at will—a computational experiment—than to initialize a new laboratory experiment. Many new issues arise, notably noise in the form of fluctuations, from the microscopic solver. The conceptual point, however, remains: even if we do not have the right macroscopic equation for the concentration, we can still perform its numerical integration without obtaining it in closed form. The skeleton of the wrapper (the integration algorithm) is the same one we would use if we had the macroscopic equation, but now function evaluations are substituted by short computational experiments with the microscopic simulator, whose results are appropriately processed for local macroscopic identification and estimation. If a large separation of time scales exists between microscopic dynamics (here, the time we need to run kMC to estimate dc/dt) and the macroscopic evolution of the concentration, this procedure may be significantly more economical than direct simulation.
Passing information between the microscopic and macroscopic scales at the beginning and the end of each computational experiment is vitally important. This is accomplished through a lifting operator (macro- to micro-) and a restriction operator (micro- to macro-) as discussed below (Theodoropoulos et al., 2000; Kevrekidis et al., 2003 and references therein). The proposed computational methodology consists of the following basic elements:
Choose the statistics of interest for describing the long-term behavior of the system and an appropriate representation for them. For example, in a gas simulation at the particle level, the statistics would probably be density and momentum (zeroth and first moment of the particle distribution over velocities), and we might choose to discretize them in a computational domain via finite elements. We call this the macroscopic description, u. These choices suggest possible restriction operators, M, from the microscopic-level description U, to the macroscopic description: u = MU.
Choose an appropriate lifting operator, μ, from the macroscopic description, u, to one or more consistent microscopic descriptions, U. For example, in a gas simulation using pressure etc. as the macroscopic-level variables, μ could make random particle assignments consistent with the macroscopic statistics, μM=I (i.e., lifting from the macroscopic to the microscopic and then restricting [projecting] down again should have no effect, except roundoff).
Start with a macroscopic condition (e.g., concentration profile) u(t0).
Transform it through lifting to one or more fine, consistent microscopic realizations U(t0) = μu(t0).

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Evolve this(these) realization(s) using the microscopic simulator for the desired short macroscopic time T, generating the value(s) U(T);
Obtain the restriction(s) u(T)=MU(T) (and average over them).
This constitutes the coarse time-stepper, or coarse time-T map. If this map is accurate enough, it can be used as I described in a two-tier procedure to perform coarse projective integration (Gear and Kevrekidis, 2003; Gear, 2001; Gear et al., 2002) (see Figure 1a). Coarse projective integration, and also coarse bifurcation computations (see Figure 1b), have been used to accelerate lattice kMC simulations of catalytic surface reactions (Makeev et al., 2002a,b; Rico-Martinez et al., 2004), Brownian dynamics simulations of nematic liquid crystals (Siettos et al., 2003), and much more.
Time-stepper-based methods are, in effect, alternative ensembles for performing microscopic (molecular dynamics, kMC, Brownian dynamics) simulations. Innovative multiscale/multilevel techniques proposed over the last decade that can be integrated in an equation-free, time-stepper-based framework include the quasi-continuum methods of Phillips and coworkers (Phillips, 2001; Ortiz and Phillips, 1999) and the optimal prediction methods of Chorin and coworkers (Chorin et al., 1998, 2000) (see the discussion in Kevrekidis et al., 2003).
If one has good macroscopic equations, one should use them. But when these equations are not available in closed form, and such cases arise with increasing frequency in contemporary modeling, the equation-free computational enabling technology we have outlined here may hold the key to engineering effectively simple systems.
ACKNOWLEDGMENTS
This work was partially supported over the years by AFOSR, through an NSF/ITR grant, DARPA, and Princeton University. An extended version of this article appeared as a “Perspective” in the July 2004 issue of the AIChE Journal (with C. W. Gear and G. Hummer as coauthors).
REFERENCES
Box, G.E.P., W. Hunter, and J.S. Hunter. 1978. Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building. Indianapolis, Ind.: Wiley-Interscience.
Chorin, A., A. Kast, and R. Kupferman. 1998. Optimal prediction for underresolved dynamics. Proceedings of the National Academy of Sciences 95(8): 4094–4098.
Chorin, A., O. Hald, and R. Kupferman. 2000. Optimal prediction and the Mori-Zwanzig representation of irreversible processes. Proceedings of the National Academy of Sciences 97(7): 2968–2973.
Cybenko, G. 1996. Just in Time Learning and Estimation. Pp. 423-434 in Identification, Adaptation and Learning: The Science of Learning Models from Data, edited by S. Bittanti and G. Picci. NATO Advanced Studies Institute (ASI) Series F153. Berlin: Springer-Verlag.

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FIGURE 1 Schematic illustrations of (a) coarse projective integration and (b) coarse time-stepper-based bifurcation computations. Source: Gear et al., 2002. Reprinted with permission.

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Gear, C.W. 2001. Projective Integration Methods for Distributions. NEC Research Institute Technical Report 2001-130. Princeton, N.J.: NEC Research Institute. Available online at: <http://www.neci.nj.nec.com/homepages/cwg/pdf.pdf>.
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