MONTE-CARLO QUASI-METHOD AND CUBATURES FOR SERENDIPIC POLINOMIALS

Abstract

The overwhelming majority of calculations by Monte-Carlo method is done with the use of pseudo-random numbers (quasi-random points). Practice has shown that in some cases it is better to refuse from modelling real random process and to use artificial model instead. Computational templates and cubatures with quasi-random nodes of integration are considered in the paper. With concrete examples of biquadratic and bicubic polynomials it is shown that sequence of quasi-random points gives better results. It is a known fact and the essence of Monte-Carlo quasi-method. The Monte Carlo quasi-method is based on a square computational template and a stratified sample of 9 applications. There are 3 ways to design the cubature according to the Newton-Cotes version (the Newton-Cotes procedure, the hierarchic procedure on the basis of nodal proportionality, quick algorithm for centered templates). The analysis of the results of cubature testing is carried out taking into account the specific properties and behavior of the serendipic surfaces at the border and in the middle of the carrier. A simple relationship was found between the mean surface application and the barycentric application (in the center of the square). The number of necessary integration nodes is reduced to one. In this case, the cubature of Newton-Cotes is more effective than the cubature of Gauss-Legendre. The algorithm of The Monte Carlo quasi-method for quick determination of per-node distribution of even volume force of serendipic elements of biquadratic and bicubic interpolations is built. There is reason to believe that Zienkiewicz was wrong as to the role of out-of-node parameters. It turned out that barycentric application determines “upon the average” the character of nonstandard serendipic surface.