A. Benabidallah, Y. Cherruault and G. Mora
In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using α‐dense curves and approximating multiple integrals…
Abstract
In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using α‐dense curves and approximating multiple integrals. Using first, periodic and regular α‐dense curves we propose a trapezoidal formula for calculating the periodic integrand obtained. Then, we consider the simple integrals as integrals with weight. We propose a method to evaluate the moments of the weight function. This allows us to build a recurrent formula for the orthogonal polynomials family and to use a Gaussian rule to estimate the simple integral. Finally, we adapt the Filon's method, consisting in evaluating the Fourier coefficients of a function, to the oscillatory integrand obtained by using reducing transformations.
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A. Benabidallah, Y. Cherruault and G. Mora
The authors have developed a new method for approximating multiple integrals. This new method is based on Alienor method which use α‐dense curves. Double and triple integrals have…
Abstract
The authors have developed a new method for approximating multiple integrals. This new method is based on Alienor method which use α‐dense curves. Double and triple integrals have been approximated by densification of the domain. In this paper, the aim is to prove by different ways the theory elaborated in earlier researches.
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A. Benabidallah, Y. Cherruault and Y. Tourbier
Some results for calculating double and triple integrals have been established, using α‐dense curves in the domain of integration. The technique of α‐dense curves has been first…
Abstract
Some results for calculating double and triple integrals have been established, using α‐dense curves in the domain of integration. The technique of α‐dense curves has been first investigated. Our aim in this paper is to give an approximation of multiple integrals in [0,1] d (d∈N*) using α‐dense curves in [0,1] d, and to evaluate the error method.
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G. Mora, Y. Cherruault, A. Benabidallah and Y. Tourbier
This paper is intended to provide a numerical method for computing integrals of several variables. The method is based on a intuitive geometric idea relative to the meaning of…
Abstract
This paper is intended to provide a numerical method for computing integrals of several variables. The method is based on a intuitive geometric idea relative to the meaning of densifying a domain in Rn+1(n≥1) by a curve h(t), contained in that domain, say K, with a very small density α (this must be interpreted as the following property: for any point of K there exists a point of the curve at distance less or equal than α).Thus, the method states that any area, volume, etc, can be computed as the limit of the length of a certain curve, densifying that domain, multiplied by a power of its density. Therefore, the computation of a multiple integral of a nonnegative continuous function can be approached by a simple integral corresponding to the length of the curve h(t) and certain power of its density.
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A. Benabidallah, Y. Cherruault and Y. Tourbier
The Alienor method, based on α ‐dense curves, has been developed by Yves Cherruault and collaborators, to solve optimization problems. It can be coupled with the decomposition…
Abstract
The Alienor method, based on α ‐dense curves, has been developed by Yves Cherruault and collaborators, to solve optimization problems. It can be coupled with the decomposition method of Adomian to solve optima control problems also. But α ‐dense curves can be used in many other problems. Gives an application of α ‐dense curves for calculating multiple integrals by means of simple integrals.
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A. Benabidallah and Y. Cherruault
To study constrained or unconstrained global optimization problems in a cube of Rd where d is a positive integer.
Abstract
Purpose
To study constrained or unconstrained global optimization problems in a cube of Rd where d is a positive integer.
Design/methodology/approach
α‐dense curves are initially used to transform this problem into a global optimization problem of a single variable. The optimization of the one variable is then treated by means of the Legendre‐Fenchel Transform. This discrete convex envelope of the one variable function obtained previously, can then be computed.
Findings
Global optimization problems of this nature have already been extensively studied by the authors. In this paper they have coupled the Alienor method with Legendre‐Fenchel Tranform to compute a discrete convex envelope of the function to minimize. A fast algorithm was successfully used to do this.
Research limitations/implications
This approach to global optimization is based on α‐dense curves and numerical tests performed on a Pentium IV (1,700 MHz) computer used with Mathematica 4 software.
Practical implications
Gives the solutions illustrated in the numerous examples provided that show the practicality of the methodology.
Originality/value
A new approach based on extensive research into global optimization via α‐dense curves.
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A. Benabidallah, Y. Cherruault and Y. Tourbier
Decomposition of several variables functions by means of functions of one variable was a fundamental problem, studied by mathematicians, and the specially by KOLMOGOROV school…
Abstract
Decomposition of several variables functions by means of functions of one variable was a fundamental problem, studied by mathematicians, and the specially by KOLMOGOROV school. This question is closely connected with optimization and optimal control and with multiple integrals calculus. These problems have been investigated by Professor Y. Cherruault and colleagues, using the ALIENOR method, which is based on α‐dense curves. The decomposition method of Adomian can be coupled with global optimization for solving optimal control problems. Aim is to calculate multiple integrals by a special decomposition of the function using an orthonormal basis of functions. Presents also new methods for approximating a n‐variables function by means of the sum of product of n functions only depending on a single variable. Applications to multi‐variables optimization problems and optimal control system are described.
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G. Mora, Y. Cherruault and A. Benabidallah
In this paper, some kinds of operators, defined on a class of real functions depending on a single real variable, preserving some fundamental properties for solving global…
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In this paper, some kinds of operators, defined on a class of real functions depending on a single real variable, preserving some fundamental properties for solving global optimization problems are introduced. These operators allow the transfer of the search of the global extremum of a determined objective function, f into those of its transformed function, Tf, for which a global extremum is easy to obtain.
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Yves Cherruault, Gaspar Mora and Yves Tourbier
Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This…
Abstract
Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This can be performed by using α‐dense curves in Rn, already introduced for global optimization using the ALIENOR method.