Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

2020 ◽  
Vol 31 (12) ◽  
pp. 2050172
Author(s):  
Henryk Fukś ◽  
Yucen Jin
Keyword(s):  
Dynamical System ◽  
Local Structure ◽  
Structure Theory ◽  
Dimensional System ◽  
Perfect Agreement ◽  
Infinite Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Dimensional Dynamical System

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

scholarly journals MCMC METHODS FOR DIFFUSION BRIDGES

2008 ◽  
Vol 08 (03) ◽  
pp. 319-350 ◽  
Author(s):  
ALEXANDROS BESKOS ◽  
GARETH ROBERTS ◽  
ANDREW STUART ◽  
JOCHEN VOSS
Keyword(s):  
Mcmc Methods ◽  
Recent Theory ◽  
The Novel ◽  
Infinite Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
It Diffusion ◽  
Dimensional Approximation ◽  
Independence Sampler ◽  
Random Walk Metropolis Algorithm

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

scholarly journals K-adaptability in two-stage mixed-integer robust optimization

2019 ◽  
Vol 12 (2) ◽  
pp. 193-224 ◽  
Author(s):  
Anirudh Subramanyam ◽  
Chrysanthos E. Gounaris ◽  
Wolfram Wiesemann
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Finite Convergence ◽  
Two Stage ◽  
Infinite Dimensional ◽  
Benchmark Data ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Abstract We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a K-adaptability formulation that selects K candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.

Center Manifold of Fractional Dynamical System

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Dynamical System ◽  
Center Manifold ◽  
High Dimensional ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Reduction Methods ◽  
Fractional Dynamical System ◽  
Lower Dimensional

Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.

scholarly journals Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2495
Author(s):  
Alexandre Mauroy
Keyword(s):  
Linear Method ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Convergence Results ◽  
Dimensional Approximation

We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from the data, a method which can be seen as a generalization of the Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.

scholarly journals INVARIANT MEASURES OF STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS USING FOURIER APPROXIMATIONS

2011 ◽  
Vol 21 (01) ◽  
pp. 113-123 ◽  
Author(s):  
MD SHAFIQUL ISLAM ◽  
PAWEL GÓRA
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Invariant Measures ◽  
Invariant Density ◽  
Stochastic Perturbations ◽  
Fourier Approximation ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Frobenius Operator

We consider dynamical system τ : [0, 1] → [0, 1] and its stochastic perturbations [Formula: see text], N ≥ 1. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let [Formula: see text] be an invariant density of τ and [Formula: see text] be a fixed point of PN. We show that [Formula: see text] converges in L1 to [Formula: see text].

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