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43872-AC9
Equation-Free, Coarse-Grained Computation for Multiphase Flow Modeling

Ioannis G. Kevrekidis, Princeton University

The purpose of this research is to accelerate the way coarse-grained, macroscopic results are obtained through fine-scale (atomistic, molecular, individual-based) code – in particular, to accelerate molecular dynamics-based computations of granular / gas-solid flows. The tool of choice is the so-called equation-free approach, which assumes the existence of coarse-grained, macroscopic conservation equations (e.g. in terms of particle volume fraction fields) which, however are not available in closed form. The approach circumvents the derivation of explicit closures by performing appropriately initialized short bursts of fine-scale simulation, processing of their results, and use of the obtained quantities to design new computational experiments with the fine scale code. An important development for last year involved the acceleration of coarse-grained demixing (segregation) of mixtures of two different types of particles (same size, different density) in a narrow fluidized bed (see Nugget). This work involved two different types of particles; we have also worked towards the development of equation-free versions of QMOM and DQMOM algorithms for the solution of integral equations that involve interactions of particles with distributed features (e.g. sizes).

In our previous work we assumed that the appropriate coarse-grained variables, that is, the variables in terms of which the (explicitly unavailable) closed macroscopic conservation equations would be written, are known in advance. In our coarse segregation computations, for example, we used a couple of different discretizations of the inverse cumulative particle distribution function along the bed as the coarse description. In most complex systems of contemporary interest, however, we do not a-priori know how to find such important emergent-level variables. We are thus faced with the task of extracting the emergent level variables from fine scale data. This year we concentrated on data mining tools that would allow us to extract good coarse observables from the fine scale simulation in a computer-automated way. A new tool for this kind of nonlinear data analysis is harmonic analysis on maps, and in particular, a “computational enabling technology” termed “diffusion maps” [1].

This is a data-mining technique (in the general class of the so-called “manifold learning techniques”) whose purpose is, starting with large ensembles of high-dimensional data (points in Rⁿ with n >> 1), to extract low-dimensional representations of the data. One can think of this approach as the nonlinear extension of classical Principal Component Analysis. We briefly describe how the procedure works when the data are long vectors, such as the ones that would result from describing the positions of particles in a snapshot of a granular computation. Consider, as an illustration, an ensemble of three-dimensional data (like the so-called “swiss-roll” data set, lying on a two dimensional nonlinear manifold in R³; the approach “realizes” algorithmically the effective two-dimensionality of the data, finally “unfolding” them in R² – and in the process discovering the coordinates in which this unfolding and reduction can be done. For “very nearby” data points (“snapshots” of the system state, vectors), the Euclidean distance is a good measure of the ease with which a change or transition can be made from one snapshot to the other; but when this Euclidean distance is greater than a cutoff, say s, then the Euclidean distance is not a good measure of “transition difficulty” any more. Diffusion maps provide a way of computing a diffusion distance: a distance that is constructed based on local Euclidean distances, and in which states that are equally easy to get to from a given state lie on the surface of a sphere. The computation is based on the construction of a Markovian matrix M based on the data and a diffusion kernel that includes the parameter s. The leading eigenvalues and eigenvectors of this matrix can be computed (possibly through fast, O(NlogN), algorithms). There is always a trivial eigenvalue at 1; if a gap prevails between the next few eigenvalues and the remaining “many”, then the first few, leading eigenvalues of this matrix are “the right coordinates” leading to data reduction – they parameterize a low-dimensional manifold close to which the data lie. In effect, these are the eigenfunctions of the Laplacian on a graph whose vertices are the data, and whose edges are “decorated” with conductivities depending as implied above on local Euclidean distances. These “conductivities” become effectively zero when the Euclidean distance is larger than the cutoff. To summarize: good reduced coordinates for the coarse-grained description of a system can be constructed by computing the leading eigenfunctions of the graph Laplacian, on a graph constructed using extensive ensembles of simulation data from the problem; the crucial quantities used are local Euclidean distances between nearby data points.

[1] R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner and S. Zucker: Geometric diffusion as a tool for harmonic analysis and structure definition of data: Diffusion maps. PNAS 102, 7432-7437 (2005).

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