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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.

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.

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.

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.

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.

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.

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.

A recursive scheme for the ALIENOR method is proposed as a remedy for the difficulties induced by the method. A progressive focusing on the most promising region, in combination with a variation of the density of the alpha-dense curve, is proposed.

ALIENOR method is aimed at reducing the space dimensions of an optimization problem by spanning it by using a single alpha-dense curve: the curvilinear abscissa along the curve becomes the only design parameter for any design space. As a counterpart, the transformation of the objective function in the projected space is much more difficult to tackle.

A fine tuning of the procedure has been performed in order to identity the correct balance between the different elements of the procedure. The proposed approach has been tested by using a set of algebraic functions with up to 1,024 design variables, demonstrating the ability of the method in solving large scale optimization problem. Also an industrial application is presented.

In the knowledge of the author there is not a similar paper in the current literature.

To propose a new reducing transformation that allows the calculation of the absolute minimum of a function that depends on a large number of variables to be performed quickly.

The methodology depends on the building of α‐dense curve in a compact of Rⁿ that allows the approximate at any point of this compact with a desired precision. This approach allows global optimization problems that depend on a large number of variables, be tackled quickly and precisely.

It was found that this new method for densifying a space Rⁿ (compact) by means of simple parametric curve (a space filling curve) could be used to deal with global optimization problems of some several hundreds or thousands of variables in some seconds or minutes. The technique is being based on the cosine function.

The results depend on the use of a computer system or “micro‐calculator”.

This is an “economic” method and the technique which uses the cosine function allows the reduction of the calculation time and avoids calculus errors. It has the practical advantages that coupled with a transformation eliminating local minima, it permits the solution of global optimization problems of more than 1,000 variables in less than 1 min.

The method is innovative and shown to be accurate and fast even with a function of a large number of variables.

This paper seeks to present an original method for transforming multiple integrals into simple integrals.

This can be done by using α‐dense curves invented by Y. Cherruault and A. Guillez at the beginning of the 1980s.

These curves allow one to approximate the space Rⁿ (or a compact of Rⁿ) with the accuracy α. They generalize fractal curves of Mandelbrobdt. They can be applied to global optimization where the multivariables functional is transformed into a functional depending on a single variable.

Applied to a multiple integral, the α‐dense curves using Chebyshev's kernels permit one to obtain a simple integral approximating the multiple integral. The accuracy depends on the choice of α.

The paper presents an original method for transforming integrals into simple integrals.