Pseudo-differential Operators, Cubature and Equidistribution on the 3D ball: An Approach Based on Orthonormal Basis Systems. (3rd July 2017)

• Record Type: Journal Article
• Authors: Ishtiaq, Amna; Michel, Volker
• Is Part Of: Numerical functional analysis and optimization. Volume 38:Number 7(2017)
• Page Start: 891
• Page End: 910
• Publication Date: 2017-07-03
• Subjects: Discrepancy -- numerical integration -- point grids -- unit ball -- uniformity measure; 46E35 -- 47G30 -- 65D30 -- 65D32; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1293092
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this paper, the distribution of points on a unit ball in ℝ 3 is investigated. The ansatz is motivated by an approach for point grids on the unit sphere by Cui and Freeden. A formula for a generalized discrepancy is developed, which is then used to check the uniformity of point grids on a ball. The generalized discrepancy originates from an error bound for a quadrature (cubature) rule on the ball with uniform weights. In particular, we discuss the integration of functions from particular Sobolev spaces based on known orthonormal systems on the ball. This includes the introduction of a concept of pseudo-differential operators on the ball. Finally, different point grids are constructed on the ball and are compared by the discrepancy. Furthermore, numerical and graphical comparisons of the grids are presented.