OPTIMAL METHODS OF PHYSICAL FIELD TABULATION 

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), boikov@pnzgu.ru
Boykova Alla Il'inichna, Candidate of physical and mathematical sciences, senior staff scientist, sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), math@pnzgu.ru
Krivulin Nikolay Petrovich, Candidate of engineering sciences, associate professor, sub-department of higher
and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), krivulin@bk.ru
Grinchenkov Grigoriy Igorevich, Postgraduate student, Penza State University (40 Krasnaya street, Penza,
Russia), ryazantsevv@mail.ru 

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Background. Physical fields of various nature, as a rule, have a complex structure that is impossible to be represented analytically. Solution of many problems of physics and engineering requires high precision representation of information on on both constant and variable physical fields. Under the notion of a physical field in a certain range we understand uniform approximation of a field with given accuracy in a range under investigation. Standard methods, based on field approximation by equidistant nodes, lead to significant errors. Therefore the problems of formation of uniform (in one or another metrics) approximation of fields in the given range are topical ones. The second topical problem is the development of optimal methods of information tabulation and transfer enabling to restore a field with the given accuracy. The article is devoted to solution the said problems.
Materials and methods. To solve the said problems the authors suggest a method common for physical fields of any nature, which is based as follows: 1) building of algorithms of uniform approximation of fields in the range under investigation; 2) development of optimal methods of field information tabulation; 3) building of an apparatus of table decoding, enabling to restore a physical field in the given range with predetermined accuracy. In order to form thebest uniform approximation of a physical field it is necessary to determine a functional class, which the said field belong to, to calculate Kolmogorov widths of the corresponding function class and to build splines being an optimal method of approximation. After that, a physical field is tabulated using the information on the function class. Tabulation of physical fields is naturally based on the concept of Kolmogorov entropy. The last step is the development of an apparatus of restoration of a physical field with the given accuracy on the basis of tabulation results.
Results. The authors suggest the methods, optimal in accuracy and memory, of restoration of potential fields of various nature: of Newtonian and Coulomb potentials, of electrostatic fields.
Conclusions. The results of the study may be applied in development of optimal methods of acquisition and transfer of information on physical fields of various nature.

physical field, uniform approximation, optimal methods of information tabulation, width, spline. 

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