Interpolative Decomposition Butterfly Factorization

Pang, Qiyuan; Ho, Kenneth L.; Yang, Haizhao

doi:10.48550/arxiv.1809.10573

Cited by 2 publications

(14 citation statements)

References 27 publications

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“…The notation of the 1D IDBF introduced in [27] will be adopted and adjusted to the multidimensional case in this paper. With no loss of generality, we focus on the 2D case with uniform point distributions first.…”

Section: Notations and Overall Structurementioning

confidence: 99%

“…With no loss of generality, we focus on the 2D case with uniform point distributions first. The notations and overall structure discussed below are similar to that in [21,27].…”

Section: Notations and Overall Structurementioning

confidence: 99%

“…When the point sets are non-uniform, the non-uniform FFT (NUFFT) algorithms in [12,31] are able to evaluate (1) when Φ(x, ξ) = x • ξ with O(N log(N )) operations. For more general kernel functions, the butterfly factorization (BF) [20,24,26,27] can factorize the dense matrix e 2πiΦ(x,ξ) as a product of O(log(N )) sparse matrices, each of which has only O(N ) non-zero entries. Hence, storing and applying e 2πiΦ(x,ξ) via the BF for evaluating (1) take only O(N log(N )) complexity.…”

Section: Introductionmentioning

confidence: 99%

“…However, for multidimensional kernel functions, existing algorithms are efficient only when the explicit formula of the phase function Φ is known [1,31,24,6,19,20,22,27]. The computational challenge in the case of indirect access of the kernel function (see Table 1 for a list of different scenarios) motivates a series of new algorithms for multidimensional cases in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…Secondly, when lowrank phase matrices have been recovered, a new BF, multidimensional interpolative decomposition butterfly factorization (MIDBF), is proposed for the matvec Kf with O(N log N ) complexity for both precomputation and application. The MIDBF is a generalization of the interpolative decomposition butterfly factorization (IDBF) [27] in multidimensional cases especially when the discretization grid points are non-uniform. These two contributions lead to the first framework for multidimensional fast oscillatory integral transforms in the case of indirect access with non-uniform grid points.…”

Section: Introductionmentioning

confidence: 99%

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Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization

Chen,

Zhang,

et al. 2019

Preprint

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This paper focuses on the fast evaluation of the matvec g = Kf for K ∈ C N ×N , which is the discretization of a multidimensional oscillatory integral transform g(x) = K(x, ξ)f (ξ)dξ with a kernel function K(x, ξ) = e 2πiΦ(x,ξ) , where Φ(x, ξ) is a piecewise smooth phase function with x and ξ in R d for d = 2 or 3. A new framework is introduced to compute Kf with O(N log N ) time and memory complexity in the case that only indirect access to the phase function Φ is available. This framework consists of two main steps: 1) an O(N log N ) algorithm for recovering the multidimensional phase function Φ from indirect access is proposed; 2) a multidimensional interpolative decomposition butterfly factorization (MIDBF) is designed to evaluate the matvec Kf with an O(N log N ) complexity once Φ is available. Numerical results are provided to demonstrate the effectiveness of the proposed framework.

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Section: Notations and Overall Structurementioning

confidence: 99%

“…With no loss of generality, we focus on the 2D case with uniform point distributions first. The notations and overall structure discussed below are similar to that in [21,27].…”

Section: Notations and Overall Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization

Chen,

Zhang,

et al. 2019

Preprint

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A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces

Liu

Yang

2020

Journal of Computational Physics

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This paper introduces a hierarchical interpolative decomposition butterfly-LU factorization (H-IDBF-LU) preconditioner for solving two-dimensional electric-field integral equations (EFIEs) in electromagnetic scattering problems of perfect electrically conducting objects with open surfaces. H-IDBF-LU leverages the interpolative decomposition butterfly factorization (IDBF) to compress dense blocks of the discretized EFIE operator to expedite its application; this compressed operator also serves as an approximate LU factorization of the EFIE operator leading to an efficient preconditioner in iterative solvers. Both the memory requirement and computational cost of the H-IDBF-LU solver scale as O(N log 2 N ) in one iteration; the total number of iterations required for a reasonably good accuracy scales as O(1) to O(log 2 N ) in all of our numerical tests. The efficacy and accuracy of the proposed preconditioned iterative solver are demonstrated via its application to a broad range of scatterers involving up to 100 million unknowns.

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Interpolative Decomposition Butterfly Factorization

Cited by 2 publications

References 27 publications

Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization

Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization

A hierarchical butterfly LU preconditioner for two-dimensional electromagnetic scattering problems involving open surfaces

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