Abstract. 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.Key words. Model reduction, slow invariant manifold, optimization, calculus of variations, extremum principle, curvature, chemical kinetics AMS subject classifications. 37N40, 37M99, 80A30, 92E201. Introduction. In dissipative ordinary differential equation systems modeling chemical reaction kinetics the phase flow generally causes anisotropic volume contraction due to multiple time scales with spectral gaps. This leads to a bundling of trajectories near "invariant manifolds of slow motion" of successively lower dimension during time evolution. Model reduction methods exploit this for simplifying the underlying ordinary differential equation models via time scale separation into fast and slow modes and eliminating the fast modes by enslaving them to the slow ones as a graph of a function which defines the slow invariant (attracting) manifold (SIM).Early model reduction approaches in chemical kinetics like the quasi steady-state and partial equilibrium assumption [32] have been performed "by hand", modern numerical approaches are supposed to automatically compute a reduced model without need for detailed expert knowledge of chemical kinetics by the user. Many of these techniques are based on an explicit time-scale analysis of the underlying ordinary differential equation (ODE) system. Among those methods that became popular in applications are the intrinsic low dimensional manifold (ILDM) method [21] and recent extensions of its main ideas, e.g. the global quasi-linearization (GQL) [4], computational singular perturbation (CSP) [15,16], Fraser's algorithm [6,9,23], the method of invariant grids [5,11,12], the constrained runs algorithm [10,33], rate-controlled constrained equilibrium (RCCE) [14], the invariant constrained equilibrium edge preimage curve (ICE-PIC) method [27,28], flamelet-generated manifolds [7,30], and finite time Lyapunov exponents [22]. For a comprehensive overview see e.g. [11] and references therein.Reaction trajectories in phase space that are solutions of an ODE systemẋ(t) = f (x(t)), x(0) = x 0 , f ∈ C ∞ , describing chemical kinetics are uniquely det...

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.

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Model reduction methods often aim at an identification of slow invariant manifolds in the state space of dynamical systems modeled by ordinary differential equations. We present a predictor corrector method for a fast solution of an optimization problem the solution of which is supposed to approximate points on slow invariant manifolds. The corrector method is either an interior point method or a generalized Gauss-Newton method. The predictor is an Euler prediction based on the parameter sensitivities of the optimization problem. The benefit of a step size strategy in the predictor corrector scheme is shown for an example.

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