Abstract
Cubature schemes, in this case for uniformly weighted integration over the unit disk, enable exact evaluation of numerical integrals of polynomials but have been explicitly constructed for only low or moderate degrees. In this paper, cubature formulae are discovered for a wider range of degrees by leveraging numerical optimization. These results include a degree-17 cubature scheme with fewer points than existing solutions and up to a degree-77 solution with 1021 cubature points. Optimization heuristics and patterns in the distributions of cubature points are discussed, which serve as vital guides in this work. For example, these heuristics leverage a connection to circle-packing configurations to facilitate the discovery of fully symmetric cubature schemes.
Original language | English |
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Article number | 114076 |
Pages (from-to) | 1-19 |
Number of pages | 19 |
Journal | Journal of Computational and Applied Mathematics |
Volume | 407 |
DOIs | |
Publication status | Published - Jun 2022 |
Bibliographical note
© 2022 The Authors. Published by Elsevier B.V. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.Keywords
- Cubature formulas
- Numerical integration
- Unit disk
- Least-squares optimization