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arXiv · 1509.07925

Multidimensional Butterfly Factorization

Yingzhou Li·Haizhao Yang·Lexing Ying
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Abstract

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 $N\times N$ with a product of $Ø(\log N)$ sparse matrices, each of which contains $Ø(N)$ 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 $Ø(1)$ 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.

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Yingzhou Li, Haizhao Yang, Lexing Ying. 2015-09-25. Multidimensional Butterfly Factorization. https://doi.org/10.1016/j.acha.2017.04.002

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