PROBABILITY: FROM HERMITE POLYNOMIALS TO GAUSS QUADRATURE - Khomchenko, A. N.; Lytvynenko, O. I.; Astionenko, I. O. (Computer Science and Applied Mathematics, 2021)
The article is devoted to the use of probabilistic models in improbabilistic problems. The new examples presented in the paper will help to increase the number of supporters of randomization in mathematical modeling. Problems of restoration of finite functions (“lids”-functions, Hermite functions), which are widespread in a method of finite elements (MFE) are considered in the work. The “lid”-function is another name for the barycentric coordinate proposed by Möbius. In contrast to Lagrange interpolation, Hermite interpolation provides for the presence of information about a function and its derivative at the apexes of the control interval. Increasing Hermite polynomials at canonical intervals [0; 1] and [-1; 1] are considered as functions of probability distribution. Two methods of constructing Hermite polynomials are compared: traditional (matrix) and non-traditional (probabilistic). It is shown that the density and average quadratic deviation of the Hermite probability distribution law are closely related to the formulas of approximate integration (quadratures) of enhanced accuracy: Gauss-Bernoulli (two nodes on [0; 1]), Gauss-Legendre (two nodes on [- 1; 1]), Gauss-Lobatto (for four nodes). These results indicate the presence of a “reverse movement” of ideas and methods from probability theory to other mathematical sciences. A prominent Ukrainian scientist, a specialist in probability theory and random processes, the academician A.V. Skorokhod has repeatedly drawn attention to the urgent need for “reverse movement”. It is very important that the “reverse movement” was supported by all mathematicians, both “probabilists” and “non-probabilists” (A.V. Skorokhod’s term). It is not the first time that the obtained results convince that geometric probability is a simple, clear and very effective method of mathematical modeling. Not surprisingly modern information technologies begin with cognitive models of applied geometry. Such models are usually mathematically sound and physically adequate.