Optimal methods for approximate recovery of functions and solutions of partial differential equations bycomputational units by linear combinations of Korobov grids with information supercompression andrelated issues

Abstract

An amazing example of information supercompression is Korobov grids (points in the Euclidean space ofarbitrary dimensions), which are determined by two positive integers, one of which is the number of nodes. As it turnedout, quadrature formulas with equal weights and Korobov grid are almost optimal in the problem of numerical integration,while in problems of recovery of functions, at least, square times worse than optimal (Theorem 2).Thus, the following question arises "Is it possible, if possible, how to use the highest quality of Korobov meshes inrecovery problems", he himself received a positive answer in the problems of restoration of functions and transformationsof their multiple trigonometric Fourier series, in particular, containing solutions of equations in partial derivatives, inclasses of functions that are difficult in Computational Mathematics, Numerical Analysis and Approximation Theory withdominating mixed derivative and difference. Namely, grids were constructed that are explicitly linear combinations of theoriginal Korobov grids, which retain the properties of their overcompression in computational practice (Theorems 3, 5, 6,and 8).In the process of solving these problems, various results of independent significance in discrete mathematics wereobtained, in terms of the quality of the applications incorporated in them, perhaps even no less than what was achieved with their help. If characteristic functions of one-dimensional lattices are among the well-known ones with a wide range ofapplications, then the author constructed their characteristic functions in Euclidean spaces of any dimension for arbitrarylattices with integer nondegenerate master matrices (Lemma 1.2.3).Another result in this series are explicit congruence solutions (Lemma 1.2.5) that arise in many problems of discretemathematics, among which are Linear congruential generators for constructing random numbers using Coway-MacPhersontests, which, with all the efforts in Computer Science, did not yield to solutions in for almost half a century, the course ofthe search for which was constantly covered in all editions of the monograph "The Art of Programming" by Donald Knuth,included in the list of 12 highest publications of the physical and mathematical cycle in the 20th century, with the closureof the problem in 2016 by N. Temirgaliyev.However, all these results, and by no means only, that were included in the author’s Ph.D thesis.