Geometric Criteria for Model Reduction in Chemical Kinetics via Optimization of Trajectories

Lebiedz, Dirk; Reinhardt, Volkmar; Siehr, Jochen; Unger, J.

doi:10.1007/978-3-642-14941-2_12

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…For the reverse mode it is shown analytically in [27], that the POI identifies the SIM exactly for an infinite time horizon, that is for t 0 → −∞, applied to the linear two-dimensional system (1) and the nonlinear two-dimensional Davis-Skodje test model (2). Accordingly, the SIM approximation error decreases exponentially with increasing time interval (t f fixed) which is also confirmed by numerical results for realistic chemical combustion processes including thermochemistry (see [26]).…”

Section: Trajectory Based Optimization Approachsupporting

confidence: 61%

“…In [22] and follow-up publications of Lebiedz et al [23][24][25][26][27]35] a species reconstruction method for identifying SIMs is presented based on a variational principle exploiting trajectory-based optimization min…”

Section: Trajectory Based Optimization Approachmentioning

confidence: 99%

“…Furthermore, there are other approaches, whereof a few are presented in Chapter 3 of this work, including the Invariant Constrained Equilibrium Edge PreImage Curve (ICE-PIC) introduced by Ren et al in 2006 [36,37], an iterative model reduction method called Zero-Derivative Principle (ZDP) presented in [12,46], the Flow Curvature Method proposed by Ginoux in [14], the Functional Equation Truncation (FET) approach by Roussel [39,40], and methods by Adrover et al [1,2] and Al-Khateeb et al [4]. Additionally, there is a trajectory-based optimization approach based on a variational principle proposed by Lebiedz et al, described in [22][23][24][25][26][27]35].…”

Section: Introductionmentioning

confidence: 99%

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On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Lebiedz¹,

Unger²

2016

Mathematical and Computer Modelling of Dynamical Systems

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Chemical kinetic models in terms of ordinary differential equations correspond to finite dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode-description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion of fundamental and unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We relate both of them in a variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as underlying fundamental principles and potentially unifying elements of diverse dimension reduction approaches.

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Section: Trajectory Based Optimization Approachsupporting

confidence: 61%

Section: Trajectory Based Optimization Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Lebiedz¹,

Unger²

2016

Mathematical and Computer Modelling of Dynamical Systems

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“…We also used this strategy in previous publications as e.g. [22,23]. However, this is impossible in case of pressure dependent reactions and it may lead to inconsistent values of thermodynamic quantities in the mass action kinetics in relation to the heat of reaction.…”

Section: Rtmentioning

confidence: 99%

An Optimization Approach to Kinetic Model Reduction for Combustion Chemistry

Lebiedz¹,

Siehr

2014

Flow Turbulence Combust

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Model reduction methods are relevant when the computation time of a full convection-diffusion-reaction simulation based on detailed chemical reaction mechanisms is too large. In this article, we review a model reduction approach based on optimization of trajectories and show its applicability to realistic combustion models. As most model reduction methods, it identifies points on a slow invariant manifold based on time scale separation in the dynamics of the reaction system. The numerical approximation of points on the manifold is achieved by solving a semi-infinite optimization problem, where the dynamics enter the problem as constraints. The proof of existence of a solution for an arbitrarily chosen dimension of the reduced model (slow manifold) is extended to the case of realistic combustion models including thermochemistry by considering the properties of proper maps. The model reduction approach is finally applied to three models based on realistic reaction mechanisms: 1. ozone decomposition as a small test case; 2. simplified hydrogen combustion for comparison with another model reduction method; 3. syngas combustion as a test case including all features of a detailed combustion mechanism.

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“…The work of Ginoux et al is of particular significance in the context of this paper, it supported our inspiration to work on topics presented here although we have been concerned with differential geometry ideas in the context of slow invariant manifolds since some years [32] without knowing about Ginoux's work. Our model reduction technique is based on a variational principle by using a trajectory-based optimization approach is proposed by Lebiedz et al [27,31,32,33,35,34], which is supposed to be applied to kinetic models in combustion chemistry [28,29]. The recently published work of Lebiedz and Unger [30] discusses and exploits common ideas and brings together concepts of several model reduction approaches.…”

Section: Introductionmentioning

confidence: 99%

Towards Differential Geometric Characterization of Slow Invariant Manifolds in Extended Phase Space: Sectional Curvature and Flow Invariance

Heiter¹,

Lebiedz²

2018

SIAM J. Appl. Dyn. Syst.

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Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a general viewpoint using differential geometric concepts for submanifolds to deal with slow invariant manifolds in an extended phase space. The motivation is a coordinate independent formulation of the manifold properties and its characterization problem treating the manifold as intrinsic geometric object. We formulate a computationally verifiable necessary condition for the slow invariant manifold graph stated in terms of a differential geometric view on the invariance property. Its application to example systems is illustrated. In addition, we present some ideas and investigations concerning the search for sufficient, differential geometric conditions characterizing slow invariant manifolds based on our previously developed variational principle.

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Geometric Criteria for Model Reduction in Chemical Kinetics via Optimization of Trajectories

Cited by 6 publications

References 18 publications

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

An Optimization Approach to Kinetic Model Reduction for Combustion Chemistry

Towards Differential Geometric Characterization of Slow Invariant Manifolds in Extended Phase Space: Sectional Curvature and Flow Invariance

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