On the choice of distribution densities in adaptive mesh nodes for stochastic algorithms of numerical integration and function approximation
In this paper we formulate the analog of the equidistributional principle for stochastic adaptive meshes which are used for solving the multi-dimensional problems of numerical integration. It means that the integral for density of continuous 1D random variable over the interval (𝜉 1, 𝜉2 ) between two sample values of the variable has the “universal” distribution with the density equal to the so called “hat-function”. The importance sampling technique is treated as a method for constructing an adaptive mesh for the problem of numerical integration. The discrete-stochastic approach with the “modelled” basis of the Strang - Fix approximation for numerical simulation of the corresponding adaptive mesh is proposed. We also consider the problem on the choice of the distribution density of nodes in implementation of self-organization in the T. Kohonen’s algorithm. The research is aimed at construction of adaptive meshes for solving the problem of numerical function approximation. For the smooth approximated functions we propose to use the square root of the second derivative module of the approximated function as density of adaptive nodes. The example of function for which the formulated recommendation leads to improvement of approximating properties of mesh approach is given. The most part of results is formulated in the simplest 1D-version. The survey of possible multi-dimensional versions of these results and directions of further investigations is presented in the conclusion.