  The adoption of detailed mechanisms for chemical kinetics often poses two
types of severe challenges: First, the number of degrees of freedom is large;
and second, the dynamics is characterized by widely disparate time scales. As a
result, reactive flow solvers with detailed chemistry often become intractable
even for large clusters of CPUs, especially when dealing with direct numerical
simulation (DNS) of turbulent combustion problems. This has motivated the
development of several techniques for reducing the complexity of such kinetics
models, where eventually only a few variables are considered in the development
of the simplified model. Unfortunately, no generally applicable a priori recipe
for selecting suitable parameterizations of the reduced model is available, and
the choice of slow variables often relies upon intuition and experience. We
present an automated approach to this task, consisting of three main steps.
First, the low dimensional manifold of slow motions is (approximately) sampled
by brief simulations of the detailed model, starting from a rich enough
ensemble of admissible initial conditions. Second, a global parameterization of
the manifold is obtained through the Diffusion Map (DMAP) approach, which has
recently emerged as a powerful tool in data analysis/machine learning. Finally,
a simplified model is constructed and solved on the fly in terms of the above
reduced (slow) variables. Clearly, closing this latter model requires
nontrivial interpolation calculations, enabling restriction (mapping from the
full ambient space to the reduced one) and lifting (mapping from the reduced
space to the ambient one). This is a key step in our approach, and a variety of
interpolation schemes are reported and compared. The scope of the proposed
procedure is presented and discussed by means of an illustrative combustion
example.
