Interpolative Butterfly Factorization

Li, Yingzhou; Yang, Haizhao

doi:10.1137/16m1074941

Cited by 30 publications

(39 citation statements)

References 37 publications

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“…Proof. The proof of this theorem follows the same line of argument in [2,29,20] and below we outline the key idea. Denote the center of D i by (r i , s i ) and the center of X j by x j .…”

Section: 2mentioning

confidence: 99%

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SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

Khoo¹,

Ying²

2019

SIAM J. Sci. Comput.

111

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We propose a novel neural network architecture, SwitchNet, for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered field (and vice versa). The main difficulty of using a neural network for this problem is that a scatterer has a global impact on the scattered wave field, rendering typical convolutional neural network with local connections inapplicable. While it is possible to deal with such a problem using a fully connected network, the number of parameters grows quadratically with the size of the input and output data. By leveraging the inherent low-rank structure of the scattering problems and introducing a novel switching layer with sparse connections, the SwitchNet architecture uses much fewer parameters and facilitates the training process. Numerical experiments show promising accuracy in learning the forward and inverse maps between the scatterers and the scattered wave field.

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“…Proof. The proof of this theorem follows the same line of argument in [2,29,20] and below we outline the key idea. Denote the center of D i by (r i , s i ) and the center of X j by x j .…”

Section: 2mentioning

confidence: 99%

“…With the above definitions for U , V , and Σ, the approximation in (20) can be written compactly as (25) A ≈ U ΣV * .…”

Section: Matrix Factorizationmentioning

confidence: 99%

SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

Khoo¹,

Ying²

2019

SIAM J. Sci. Comput.

111

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“…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%

“…To simplify the discussion, we also assume that x and ξ are one-dimensional variables. It is easy to extend the IBF-MAT to multi-dimensional cases following the ideas in [7,18,20,21].…”

Section: Ibf-matmentioning

confidence: 99%

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

Yang

2019

Journal of Computational Physics

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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.

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An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

Bremer

2018

Adv Comput Math

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We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel's equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. We supplement these precomputed expansions with two asymptotic expansions, one for large orders and extremely small arguments and the other for large orders and extremely large arguments, and with series expansions for small orders and small arguments. Our scheme is able to evaluate Bessel functions of orders between 0 and 1, 000, 000, 000 at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has at least three advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel's differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions as well as our code for all of the numerical experiments described here is publically available.

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Interpolative Butterfly Factorization

Cited by 30 publications

References 37 publications

SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

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

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

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