Multidimensional Butterfly Factorization
arXiv:1509.07925 · doi:10.1016/j.acha.2017.04.002
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 with a product of sparse matrices, each of which contains 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 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.
References in corpus (2)
Cited by in corpus (11)
- Interpolative Butterfly Factorization
- A Unified Framework for Oscillatory Integral Transform: When to use NUFFT or Butterfly Factorization?
- Butterfly-Net: Optimal Function Representation Based on Convolutional Neural Networks
- Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion problems: corner domains
- An analysis of a butterfly algorithm
- A Hierarchical Butterfly LU Preconditioner for Two-Dimensional Electromagnetic Scattering Problems Involving Open Surfaces
- Butterfly factorization via randomized matrix-vector multiplications
- Block Basis Factorization for Scalable Kernel Matrix Evaluation
- Multidimensional Phase Recovery and Interpolative Decomposition Butterfly Factorization
- Rethinking Neural Operations for Diverse Tasks
- Approximate inversion of discrete Fourier integral operators