Schemes for cubature over the unit disk found via numerical optimization

Nick Takaki, G. W. Forbes, Jannick P. Rolland

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
59 Downloads (Pure)

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 languageEnglish
Article number114076
Pages (from-to)1-19
Number of pages19
JournalJournal of Computational and Applied Mathematics
Volume407
DOIs
Publication statusPublished - 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

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