Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics
with the small parameter 0 < τ 1 and α(u, v) and β(u, v) are O(1). For any given initial condition (u0, v0), already at t = O(τ ) the system approaches a new value (u0, v), where v satisfies the asymptotic relation β(u0, v) = 0. Although the system is fully described by two coordinates, the relation β(u, v) = 0 defines a slow one-dimensional manifold which approximates the slow dynamics for t τ . In this example it is clear that v is the fast variable while u is the slow one. Projecting onto the slow manifold here is rather easy: the fast foliation is simply “vertical”, i.e. u = const. However, when we observe the system in terms of the variables x = x(u, v) and y = y(u, v) which are unknown non-linear functions of u and v, then the “observables” x and y have both fast and slow dynamics. Projecting onto the slow manifold becomes nontrivial, because the transformation from (x, y) to (u, v) is unknown. Detecting the existence of a slow manifold under these conditions and projecting onto it are important in any model reduction technique. Knowledge of a good parametrization of such a slow manifold is a crucial component of the equation-free framework for modeling and computation of complex/multiscale systems [1, 2, 3]. In recent years, diffusion maps [4, 5, 6, 7, 8, 9] have been used to detect low-dimensional, nonlinear manifolds underlying high-dimensional data sets. In this paper we combine diffusion maps with recent tools from non-linear independent component analysis  to detect slow variables in highdimensional data arising from dynamic model simulations. The proposed algorithm takes into account the time dependence of the data, whereas in the diffusion map approach the time labeling of the data points is not included. We demonstrate our algorithm for stochastic simulators arising in the context of chemical/biochemical reaction modeling.