Computation of power law equilibrium measures on balls of arbitrary dimension
File(s)
Published version
Author(s)
Gutleb, Timon S
Carrillo, José A
Olver, Sheehan
Type
Journal Article
Abstract
We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in Gutleb et al. (Math Comput 9:2247–2281, 2022) for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark constrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.
Date Issued
2023-08-01
Date Acceptance
2022-11-23
Citation
Constructive Approximation, 2023, 58, pp.75-120
ISSN
0176-4276
Publisher
Springer
Start Page
75
End Page
120
Journal / Book Title
Constructive Approximation
Volume
58
Copyright Statement
© The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
http://arxiv.org/abs/2109.00843v1
Subjects
65N35, 65R20, 65K10
cs.NA
math.NA
math.NA
Notes
40 pages, 14 figures
Publication Status
Published
Date Publish Online
2022-12-15