Smoothness Effects on Numerical Integration Accuracy for Rapidly Oscillating Bivariate Functions on Sparse Grids

• Date: 2025
• Main Authors: Нечуйвітер, Олеся, Іванов, Владислав, Шніцар, Андрій, Гіщак, Остап
• Format: Article
• Language: English
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/347682
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

One of the key tasks in modern applied mathematics, without which it is impossible to model and analyze complex processes, particularly in digital image processing, is the numerical integration of functions of several variables. The main problem of numerical integration of functions of several variables is the increase in computational costs with increasing dimension of the integration domain. Of particular interest are numerical integration methods developed using information operators that restore intermediate values of quantities based on a given set of known values of a function of several variables at points, lines, planes, etc. Based on such operators, economical schemes for interpolating functions of several variables are constructed. The use of economical schemes in the numerical integration of functions of two and three variables allows one to construct sparse grids and calculate approximate integrals with less data and with a predetermined accuracy compared to classical methods. The purpose of this article is to demonstrate the use of economical interpolation schemes for approximate calculation of double integrals of rapidly oscillating functions of general form on different classes of smoothness. The paper analyzes the influence of the order of differentiability of a function on the rate of decay of the theoretical error of approximation of cubature formulas. It is shown that as the smoothness of the function increases, the estimates of the error of numerical integration improve, which allows the effective use of sparse grids without loss of accuracy. The results obtained establish a quantitative relationship between the class of differentiability of a function, the discretization parameters, and the frequency of oscillations, and can be used to justify the choice of numerical methods for integrating rapidly oscillating functions of two variables.