Multidimensional butterfly factorization


Journal Article (Letter)

© 2017 Elsevier Inc. This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size N×N with a product of O(log⁡N) sparse matrices, each of which contains O(N) nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in O(1) operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.

Full Text

Duke Authors

Cited Authors

  • Li, Y; Yang, H; Ying, L

Published Date

  • May 1, 2018

Published In

Volume / Issue

  • 44 / 3

Start / End Page

  • 737 - 758

Electronic International Standard Serial Number (EISSN)

  • 1096-603X

International Standard Serial Number (ISSN)

  • 1063-5203

Digital Object Identifier (DOI)

  • 10.1016/j.acha.2017.04.002

Citation Source

  • Scopus