Butterfly Factorization

Li, Yingzhou; Yang, Haizhao; Martin, Eileen; Ho, Kenneth L.; Ying, Lexing

doi:10.1137/15m1007173

Cited by 63 publications

(80 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Algorithms Precomputation time Application time memory Scenario 1 BF [19] O(N 1.5 ) O(N log N ) O(N log N ) Scenario 2 BF [19] O(N 1.5 log N ) O(N log N ) O(N 1.5 ) Scenario 3 BF [7,18] Table 2: Summary of existing algorithms and proposed algorithms (in bold) for the evaluation of Kf for a general kernel α(x, ξ)e 2πıΦ(x,ξ) when only indirect access of amplitude and phase is available according to different scenarios listed in Table 3.…”

Section: Scenariosmentioning

confidence: 99%

“…When the numerical rank of the phase function r is only larger than the dimension of the problem by one or two, NUFFT is usually faster than BF and hence it will be applied to compute Kf . Scenario 1 : There exists an algorithm for evaluating an arbitrary entry of the kernel matrix in O(1) operations [3,4,19,25].…”

Section: Scenariosmentioning

confidence: 99%

“…K := e 2πıU V * is a complementary low-rank matrix that has been widely studied in [10,11,19,21,24,25,32]. Let X and Ω be the row and column index sets of e 2πıU V * .…”

Section: Ibf-matmentioning

confidence: 99%

See 2 more Smart Citations

A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

Yang

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

This paper concerns the fast evaluation of the matvec g = Kf for K ∈ C N ×N , which is the discretization of an oscillatory integral transform g(x) = K(x, ξ)f (ξ)dξ with a kernel function K(x, ξ) = α(x, ξ)e 2πıΦ(x,ξ) , where α(x, ξ) is a smooth amplitude function , and Φ(x, ξ) is a piecewise smooth phase function with O(1) discontinuous points in x and ξ. A unified framework is proposed to compute Kf with O(N log N ) time and memory complexity via the non-uniform fast Fourier transform (NUFFT) or the butterfly factorization (BF), together with an O(N ) fast algorithm to determine whether NUFFT or BF is more suitable. This framework works for two cases: 1) explicit formulas for the amplitude and phase functions are known; 2) only indirect access of the amplitude and phase functions are available. Especially in the case of indirect access, our main contributions are: 1) an O(N log N ) algorithm for recovering the amplitude and phase functions is proposed based on a new low-rank matrix recovery algorithm; 2) a new stable and nearly optimal BF with amplitude and phase functions in a form of a low-rank factorization (IBF-MAT) is proposed to evaluate the matvec Kf . Numerical results are provided to demonstrate the effectiveness of the proposed framework.

show abstract

Section: Scenariosmentioning

confidence: 99%

Section: Scenariosmentioning

confidence: 99%

See 1 more Smart Citation

A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

Yang

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Given a dense matrix K ∈ C N ×N and a vector x ∈ C N , it takes O(N 2 ) operations to naively compute the vector y = Kx ∈ C N . There has been extensive research in constructing data-sparse representation of structured matrices (e.g., low-rank matrices [1,2,3,4], H matrices [5,6,7], H 2 matrics [8,9], HSS matrices [10,11], complementary low-rank matrices [12,13,14,15,16,17], FMM [18,19,20,21,22,23,24,25], directional low-rank matrices [26,27,28,29], and the combination of these matrices [30,31]) aiming for linear or nearly linear scaling matvec. In particular, this paper concerns nearly optimal matvec for complementary low-rank matrices.…”

Section: Introductionmentioning

confidence: 99%

“…An example of the LRCS in (14) of the complementary low-rank matrix A inFigure 2. Non-zero submatrices in(14) are shown in gray areas.…”

mentioning

confidence: 99%

Interpolative Decomposition Butterfly Factorization

Pang¹,

Ho²,

Yang³

2020

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

This paper introduces a "kernel-independent" interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary lowrank property. The IDBF can be constructed in O(N log N ) operations for an N × N matrix via hierarchical interpolative decompositions (IDs), if matrix entries can be sampled individually and each sample takes O(1) operations. The resulting factorization is a product of O(log N ) sparse matrices, each with O(N ) non-zero entries. Hence, it can be applied to a vector rapidly in O(N log N ) operations. IDBF is a general framework for nearly optimal fast matvec useful in a wide range of applications, e.g., special function transformation, Fourier integral operators, high-frequency wave computation. Numerical results are provided to demonstrate the effectiveness of the butterfly factorization and its construction algorithms.We will describe IDBF in detail in this section. For the sake of simplicity, we assume that N = 2 L n 0 , where L is an even integer, and n 0 = O(1) is the number of column or row indices in a leaf in the dyadic trees of row and column spaces, i.e., T X and T Ω , respectively. Let's briefly introduce the

show abstract

A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution

Jiang,

Greengard

2024

Comm Pure Appl Math

View full text Add to dashboard Cite

We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual‐space multilevel kernel‐splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.

show abstract

Butterfly Factorization

Cited by 63 publications

References 24 publications

A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

A unified framework for oscillatory integral transforms: When to use NUFFT or butterfly factorization?

Interpolative Decomposition Butterfly Factorization

A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution

Contact Info

Product

Resources

About