Constructive methods of invariant manifolds for kinetic problems

Gorban, Alexander N.; Karlin, Iliya V.; Zinovyev, Andreï

doi:10.1016/j.physrep.2004.03.006

Cited by 138 publications

(119 citation statements)

References 240 publications

(773 reference statements)

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“…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.…”

Section: Methodsmentioning

confidence: 99%

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

Lebiedz

2010

Entropy

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Chemical kinetic systems are modeled by dissipative ordinary differential equations involving multiple time scales. These lead to a phase flow generating anisotropic volume contraction. Kinetic model reduction methods generally exploit time scale separation into fast and slow modes, which leads to the occurrence of low-dimensional slow invariant manifolds. The aim of this paper is to review and discuss a computational optimization approach for the numerical approximation of slow attracting manifolds based on entropy-related and geometric extremum principles for reaction trajectories.

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Section: Methodsmentioning

confidence: 99%

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

Lebiedz

2010

Entropy

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“…We should mention here also growing lump and growing flag strategies used in physical and chemical applications [10], [11]. In growing lump strategy we add new nodes uniformly at the boundary of the grid using a linear extrapolation of the grid embedding.…”

Section: Adaptive Strategiesmentioning

confidence: 99%

“…To avoid this effect, we introduced in the elmap package the possibility to make a linear extrapolation of the bounded rectangular manifold (extending it by continuity in different directions). Other, more complicated extrapolations can be performed as well, like using Carleman's formulas (see [1], [5], [10], [11], [14], [15]). …”

Section: Projectingmentioning

confidence: 99%

Elastic Principal Graphs and Manifolds and their Practical Applications

Gorban

Zinovyev

2005

Computing

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Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through "the middle" of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.

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“…If ∆ is not small (in comparison with the typical value of J), then the ansatz should be improved (for details see, for example, [28,46]). It is possible to use ∆ for error estimation and correction of an ansatz after solution of projected equations too (it is so-called post-processing [47,16]). Let Ψ 0 (t), (t ∈ [0, T ]) be the solution of projected equations dΨ(t)/dt = P Ψ (J(Ψ)), and ∆(t) = J(Ψ 0 (t)) − P Ψ 0 (t) (J(Ψ 0 (t))).…”

Section: How To Evaluate the Ansatz?mentioning

confidence: 99%

“…There are different physically motivated ways to select the scalar product and create the symmetrization [14,15,16]. But symmetrization does not provide termodinamicity and the entropy for the projected equations can decrease.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban

Karlin

2004

Physica A: Statistical Mechanics and its Applications

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Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given ansatz manifold which is not tangent to the Lyapunov function levels.Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.)In example, we apply the thermodynamic projector to derivation the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally, as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2 m , and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of "molecular individualism". This is the third result.

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Constructive methods of invariant manifolds for kinetic problems

Cited by 138 publications

References 240 publications

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

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

Elastic Principal Graphs and Manifolds and their Practical Applications

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

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