Multidimensional butterfly factorization
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×N with a product of O(logN) sparse matrices, each o...
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| Veröffentlicht in: | Applied and computational harmonic analysis 2018-05, Vol.44 (3), p.737-758 |
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| container_title | Applied and computational harmonic analysis |
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| creator | Li, Yingzhou Yang, Haizhao Ying, Lexing |
| description | 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×N with a product of O(logN) sparse matrices, each of which contains O(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 O(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. |
| doi_str_mv | 10.1016/j.acha.2017.04.002 |
| format | Article |
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This factorization approximates such a kernel matrix of size N×N with a product of O(logN) sparse matrices, each of which contains O(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 O(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. 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This factorization approximates such a kernel matrix of size N×N with a product of O(logN) sparse matrices, each of which contains O(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 O(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. 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Numerical results are provided to demonstrate the efficiency of the proposed algorithms.</abstract><cop>United States</cop><pub>Elsevier Inc</pub><doi>10.1016/j.acha.2017.04.002</doi><tpages>22</tpages><orcidid>https://orcid.org/0000-0003-1852-3750</orcidid><orcidid>https://orcid.org/0000000318523750</orcidid><oa>free_for_read</oa></addata></record> |
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| source | Access via ScienceDirect (Elsevier); EZB-FREE-00999 freely available EZB journals |
| subjects | Butterfly algorithm Data-sparse matrix factorization Fourier integral operators Mathematics Operator compression Randomized algorithm |
| title | Multidimensional butterfly factorization |
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