Reduced Models in Chemical Kinetics via Nonlinear Data-Mining

Abstract

The adoption of detailed mechanisms for chemical kinetics often poses two 1 types of severe challenges: First, the number of degrees of freedom is large; and second, 2 the dynamics is characterized by widely disparate time scales. As a result, reactive flow 3 solvers with detailed chemistry often become intractable even for large clusters of CPUs, 4 especially when dealing with direct numerical simulation (DNS) of turbulent combustion 5 problems. This has motivated the development of several techniques for reducing the 6 complexity of such kinetics models, where eventually only a few variables are considered 7 in the development of the simplified model. Unfortunately, no generally applicable a priori 8 recipe for selecting suitable parameterizations of the reduced model is available, and the 9 choice of slow variables often relies upon intuition and experience. We present an automated 10 approach to this task, consisting of three main steps. First, the low dimensional manifold 11 of slow motions is (approximately) sampled by brief simulations of the detailed model, 12 starting from a rich enough ensemble of admissible initial conditions. Second, a global 13 parametrization of the manifold is obtained through the Diffusion Map (DMAP) approach, 14 which has recently emerged as a powerful tool in data analysis/machine learning. Finally, a 15 simplified model is constructed and solved on the fly in terms of the above reduced (slow) 16 variables. Clearly, closing this latter model requires nontrivial interpolation calculations, 17 enabling restriction (mapping from the full ambient space to the reduced one) and lifting 18 (mapping from the reduced space to the ambient one). This is a key step in our approach, 19 and a variety of interpolation schemes are reported and compared. The scope of the proposed 20 procedure is presented and discussed by means of an illustrative combustion example. 21 arXiv:1307.6849v1 [math.DS] 22 29the development of a plethora of approaches aiming at reducing the computational complexity of such 30 detailed combustion models, ideally by recasting them in terms of only a few new reduced variables. 31(see e.g.[1] and references therein). The implementation of many of these techniques typically involves 32 three successive steps. First, a large set of stiff ordinary differential equations (ODEs) is considered 33 for modeling the temporal evolution of a spatially homogenous mixture of chemical species under 34 specified stoichiometric and thermodynamic conditions (usually fixed total enthalpy and pressure for 35 combustion in the low Mach regime). It is well known that, due to the presence of fast and slow 36 dynamics, the above systems are characterized by low dimensional manifolds in the concentration 37 space (or phase-space), where a typical solution trajectory is initially rapidly attracted towards the 38 manifold, while afterwards it proceeds to the thermodynamic equilibrium point always remaining in 39 close proximity to the manifold. Clearly, the presence of a manifold for...

Intrinsic map dynamics exploration for uncharted effective free-energy landscapes

“…here alpha-shapes [19,20], readily implemented in the Matlab package, or more generally, "wrapping" algorithms [21]) we detect the d − 1 dimensional boundary of the region explored ("fathomed") by the available simulation data. [22,23] can also be used for this purpose.…”

“…The main assumption upon which our method is based is the same one that underpins most of the model reduction techniques in statistical mechanics: due to time scale separation, the system dynamics is mostly confined on low-dimensional (smooth) manifolds in phase-space [23,[27][28][29]. Our approach squarely aims at exploiting smoothness of the low-dimensional manifolds which, for the gradient systems of interest here, act as the support of the free energy surface governing molecular and other atomistic dynamics.…”

Equation-free analysis of agent-based models and systematic parameter determination

“…Defining the 'lifting' and 'restricting' operations is highly problem dependent [44,45,68] and necessarily have to be defined by the user of any generic code. The success or failure of the equationfree methods depend on appropriate 'lifting' and 'restricting' operation; see for example [14]. A key property required from the 'lifting' and 'restricting' operators is that if one 'lifts' and then 'restricts' without time evolving, then the original estimateX i is found [4,33,45].…”

Special Issue “Feature Papers”

“…The final two papers of the special issue address the development of such process engineering tools. In the first paper, an automated method for generating reduced order models of complex reaction systems using the approach of diffusion maps is developed and applied to an illustrative turbulent combustion problem [15]. The final paper is focused on the formulation of general iterative controller tuning as a real-time optimization problem and the application of the proposed scheme for tuning model-predictive, general fixed-order and PID controllers for both simulated and experimental systems [16].…”