arXiv subjects

Dirk Lebiedz

Publications and source records attributed to Dirk Lebiedz.

26 records · Page 2Linked to original sources

arXiv2017

An online slow manifold approach for efficient optimal control of multiple time-scale kinetics

Chemical reactions modeled by ordinary differential equations are finite-dimensional dissipative dynamical systems with multiple time-scales. They are numerically hard to tackle -- especially when they enter an optimal control problem as "infinite-dimensional" constraints. Since discretization of such problems usually results in high-dimensional nonlinear problems, model (order) reduction via slow manifold computation seems to be an attractive approach. We discuss the use of slow manifold computation methods in order to solve optimal control problems more efficiently having real-time applications in view.

math.OC↗

arXiv2014

On fundamental unifying concepts for trajectory-based slow invariant attracting manifold computation in multiscale models of chemical kinetics

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.

math.DS↗

arXiv2013

An optimization approach to kinetic model reduction for combustion chemistry

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.

physics.comp-ph↗

arXiv2013

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method.

math.OC↗

arXiv2013

A continuation method for the efficient solution of parametric optimization problems in kinetic model reduction

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.

math.OC↗

arXiv2012

A robust optimization approach to experimental design for model discrimination of dynamical systems

A high-ranking goal of interdisciplinary modeling approaches in the natural sciences are quantitative prediction of system dynamics and model based optimization. For this purpose, mathematical modeling, numerical simulation and scientific computing techniques are indispensable. Quantitative modeling closely combined with experimental investigations is required if the model is supposed to be used for sound mechanistic analysis and model predictions. Typically, before an appropriate model of a experimental system is found different hypothetical models might be reasonable and consistent with previous knowledge and available data. The parameters of the model up to an estimated confidence region are generally not known a priori. Therefore one has to incorporate possible parameter configurations of different models into a model discrimination algorithm. In this article we present a numerical algorithm which calculates a design of experiments which allows an optimal discrimination of different hypothetic candidate models of a given dynamic system for the most inappropriate parameter configurations within a parameter range via a worst case estimate. The design criterion comprises optimal measurement time points. The used criterion is derived from the Kullback-Leibler divergence. The underlying optimization problem can be classified as a semi infinite optimization problem which we solve in an outer approximation approach stabilized by a homotopy strategy. We present the theoretical framework as well as the numerical algorithmic realization.

math.OC↗

arXiv2009

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.

math.DS↗

arXiv2009

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.

math.DS↗