Coarse-graining the Dynamics of a Driven Interface in the Presence of Mobile Impurities: Effective Description via Diffusion Maps

Developing effective descriptions of the microscopic dynamics of many physical phenomena can both dramatically enhance their computational exploration and lead to a more fundamental understanding of the underlying physics. Previously, an effective description of a driven interface in the presence of mobile impurities, based on an Ising variant model and a single empirical coarse variable, was partially successful; yet it underlined the necessity of selecting additional coarse variables in certain parameter regimes. In this paper we use a data mining approach to help identify the coarse variables required. We discuss the implementation of this diffusion map approach, the selection of a similarity measure between system snapshots required in the approach, and the correspondence between empirically selected and automatically detected coarse variables. We conclude by illustrating the use of the diffusion map variables in assisting the atomistic simulations, and we discuss the translation of information between fine and coarse descriptions using lifting and restriction operators.Comment: 28 pages, 10 figure

A manifold learning approach to model reduction in combustion

We use a relatively recent nonlinear manifold learning technique (diffusion maps) to parameterize low dimensional attracting manifolds arising in the description of detailed chemical kinetics mechanisms. With no a priori knowledge about the shape and dimension of the manifold, such an approach provides a way of solving a reduced (and less stiff) set of equations in terms of automatically detected slow variables. Advantages as well as disadvantages of the approach are discussed