Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

Abstract

a b s t r a c tIn 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 four test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.

Entropy-Related Extremum Principles for Model Reduction of Dissipative Dynamical Systems

“…The resulting NLP is solved by use of the interior point algorithm implemented in the IPOPT-package [34]. As in [32] we use the invariance defect (see [35,36] and references therein) as a measure of "goodness" of the slow manifolds computed numerically. Restarting the optimization algorithm for parameter values corresponding to a point on a previously computed solution trajectory should yield the same trajectory in the case of invariance of the computed manifold.…”

“…The successive relaxation of chemical forces causes curvature in the reaction trajectories (in the sense of velocity change along the trajectory). Therefore, in [31,32] Φ…”

Reaction Kinetics Path Based on Entropy Production Rate and Its Relevance to Low-Dimensional Manifolds

“…However, for chemical reaction systems, there is no example of deriving the equation that describes a chemical change directly based on entropy production or on a Lyapunov function that does not use a chemical kinetics equation. Gorban et al [18][19][20] and Lebiedz [21][22][23] utilized the second law of thermodynamics to determine the so-called slow invariant manifold (SIM) or the so-called low-dimensional manifold (LDM). Gorban et al [18][19][20] utilized the basis orthonormal with respect to their "entropic" scalar product that uses the Hessian of a Lyapunov function (the free energy of a perfect gas in a constant volume at constant temperature) because their almost orthogonal "projector" that defines SIM helps convergence to determine SIM [20].…”

“…Gorban et al [18][19][20] utilized the basis orthonormal with respect to their "entropic" scalar product that uses the Hessian of a Lyapunov function (the free energy of a perfect gas in a constant volume at constant temperature) because their almost orthogonal "projector" that defines SIM helps convergence to determine SIM [20]. Lebiedz [21][22][23] proposed a variational principle to identify or approximate SIM as a geodesic to minimize the distance from equilibrium in concentration phase space [22,23], and they [21][22][23] proposed and compared various types of Riemannian metrics for concentration phase space. Two of these metrics are related to thermodynamics: one [22] is the Hessian of a Lyapunov function, which is the same as used by Gorban et al [18][19][20], and the other [23] is the metric of a diagonal matrix whose diagonal components correspond to each species and is defined as the sum of the products of the corresponding stoichiometric coefficients and entropy production rates of respective reactions.…”

“…Lebiedz [21][22][23] proposed a variational principle to identify or approximate SIM as a geodesic to minimize the distance from equilibrium in concentration phase space [22,23], and they [21][22][23] proposed and compared various types of Riemannian metrics for concentration phase space. Two of these metrics are related to thermodynamics: one [22] is the Hessian of a Lyapunov function, which is the same as used by Gorban et al [18][19][20], and the other [23] is the metric of a diagonal matrix whose diagonal components correspond to each species and is defined as the sum of the products of the corresponding stoichiometric coefficients and entropy production rates of respective reactions. The success of these methods in approximating SIM suggests that there might be some thermodynamic equation that determines trajectories for arriving at equilibrium in concentration phase space, although it is uncertain how the thermodynamic methods used by Gorban et al [18][19][20] and Lebiedz [21][22][23] are related to a chemical kinetics equation.…”