An Analyzable Method for Constructing a Cellular Automaton from a Continuous System

Kawaharada, Akane; Miyaji, Tomoyuki; Nakano, Nobuo

doi:10.1109/candar.2015.111

Cited by 1 publication

(8 citation statements)

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“…Recently, according to the point related to the necessity of theoretical understanding of the choice of the local rules, Kawaharada, Miyaji, and Nakano proposed a different method of collecting data in numerical experiments [4]. The method is compatible with the theoretical study on the selectivity of the CA unlike with the experimental method.…”

Section: Introductionmentioning

confidence: 54%

“…L = 10 3 is sufficiently large for constructing local rules of a deterministic CA as shown in [4]. Although a probability matrix of a stochastic CA changes depending on L, we obtained similar results for larger values of L. Section IV.B of [4] discussed the numerical study on how the resultant local rule of a CA depends on L. We used the Mersenne Twister to generate pseudorandom numbers [8].…”

Section: Numerical Examplesmentioning

confidence: 72%

“…In this section, we introduce the minimal method [4], that is a modification of the experimental method. We represent a difference equation that approximates a given PDE by…”

Section: Procedures Of the Minimal Methodsmentioning

confidence: 99%

“…These equations were chosen in their work to check the availability of the method for these simple and basic equations, where they obtained consistent results with the experimental method. The dependence on the data set size was also studied in [4] using numerical experiments. The probability matrix of a stochastic CA obtained by the minimal method converges to the limit as the size of data set L tends to infinity and that the rate of convergence is approximately 1/ √ L. On the other hand, the rule of a deterministic CA may vary, depending on the parameters of the corresponding difference equation.…”

Section: Introductionmentioning

confidence: 99%

“…In the work obtained by Kawaharada, Miyaji, and Nakano [4], the minimal method was applied to the diffusion, the advection, and Burgers' equations. These equations were chosen in their work to check the availability of the method for these simple and basic equations, where they obtained consistent results with the experimental method.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a method for constructing a cellular automaton from a continuous system

Kawaharada

Miyaji

Nakano

2016

IJNC

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A method of constructing a cellular automaton (CA) from numerical solutions of a given partial differential equation (PDE) is considered. It consists of two parts, namely, collecting spatiotemporal data numerically and finding local rules of a CA that appear most frequently.In this paper, we analyze the method mathematically to examine its selectivity and its robustness of the derived local rules so that we can ensure validity of the resultant CA model. In particular, we investigated two limit cases: (a) the number of states of CA goes to infinity and (b) the number of spatiotemporal data goes to infinity. In the former case, we prove that the resultant CA converges to the difference equation where numerical solutions of a PDE are collected. In the latter case, through mathematical analysis, we derive conditions that the resultant CA is uniquely determined when the method of constructing a CA is applied to the diffusion equation. Our study can be a theoretical foundation of empirical CA modeling methods to create a reasonable CA which can somehow reproduce the original behavior of datasets under consideration.

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Section: Introductionmentioning

confidence: 54%

Section: Numerical Examplesmentioning

confidence: 72%

“…In this section, we introduce the minimal method [4], that is a modification of the experimental method. We represent a difference equation that approximates a given PDE by…”

Section: Procedures Of the Minimal Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations