6 research outputs found
Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
In dissipative ordinary differential equation systems different time scales
cause anisotropic phase volume contraction along solution trajectories. Model
reduction methods exploit this for simplifying chemical kinetics via a time
scale separation into fast and slow modes. The aim is to approximate the system
dynamics with a dimension-reduced model after eliminating the fast modes by
enslaving them to the slow ones via computation of a slow attracting manifold.
We present a novel method for computing approximations of such manifolds using
trajectory-based optimization. We discuss Riemannian geometry concepts as a
basis for suitable optimization criteria characterizing trajectories near slow
attracting manifolds and thus provide insight into fundamental geometric
properties of multiple time scale chemical kinetics. The optimization criteria
correspond to a suitable mathematical formulation of "minimal relaxation" of
chemical forces along reaction trajectories under given constraints. We present
various geometrically motivated criteria and the results of their application
to three test case reaction mechanisms serving as examples. We demonstrate that
accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure
A variational principle for computing slow invariant manifolds in dissipative dynamical systems
A key issue in dimension reduction of dissipative dynamical systems with
spectral gaps is the identification of slow invariant manifolds. We present
theoretical and numerical results for a variational approach to the problem of
computing such manifolds for kinetic models using trajectory optimization. The
corresponding objective functional reflects a variational principle that
characterizes trajectories on, respectively near, slow invariant manifolds. For
a two-dimensional linear system and a common nonlinear test problem we show
analytically that the variational approach asymptotically identifies the exact
slow invariant manifold in the limit of both an infinite time horizon of the
variational problem with fixed spectral gap and infinite spectral gap with a
fixed finite time horizon. Numerical results for the linear and nonlinear model
problems as well as a more realistic higher-dimensional chemical reaction
mechanism are presented.Comment: 16 pages, 5 figure
Numerical optimization methods within a continuation strategy for the reduction of chemical combustion models
Model reduction methods in chemical kinetics are used for simplification of models which involve a number of different time scales. Slow invariant manifolds in chemical composition space are supposed to be identified. A selection of state variables serve for parametrization of these manifolds. Species reconstruction methods are used to compute the values of the remaining variables in dependence of the parameters. We discuss theoretical results and numerical methods for an application of a model reduction method that is developed by D. Lebiedz based on optimization of trajectories. The main focus of this work is an application of the model reduction method to models of chemical combustion. The existence of a solution of the semi-infinite optimization problem, which has to be solved to obtain a local approximation of the slow manifold, is proven. A finite optimization problem for the same purpose is presented which can be solved with a generalized Gauss-Newton method. This method is used with an active set strategy. A filter framework and iterations with second order correction are employed for globalization of convergence. Families of neighboring optimization problems can be solved efficiently in a predictor corrector continuation scheme. The tangent space of the slow manifold can be computed by evaluation of sensitivity equations for the parametric optimization problem. A step size strategy is applied in the continuation scheme for efficient progress along the homotopy path. Results of an application of the presented method are shown and discussed. The test models range from simple test examples to realistic models of syngas combustion in air