Superfast solution of linear convolutional Volterra equations using QTT approximation
We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms,...
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| Published in | Journal of computational and applied mathematics Vol. 260; pp. 434 - 448 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.04.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0377-0427 1879-1778 1879-1778 |
| DOI | 10.1016/j.cam.2013.10.025 |
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| Abstract | We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log2n). The results of the paper are illustrated by numerical examples. |
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| AbstractList | We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini's algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level to the speed of superfast Fourier transform, i.e., . The results of the paper are illustrated by numerical examples. We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log2n). The results of the paper are illustrated by numerical examples. |
| Author | Tyrtyshnikov, Eugene E. Savostyanov, Dmitry V. Roberts, Jason A. |
| Author_xml | – sequence: 1 givenname: Jason A. surname: Roberts fullname: Roberts, Jason A. email: j.roberts@chester.ac.uk organization: University of Chester, Parkgate Road, Chester, CH1 4BJ, UK – sequence: 2 givenname: Dmitry V. surname: Savostyanov fullname: Savostyanov, Dmitry V. email: dmitry.savostyanov@gmail.com organization: University of Chester, Parkgate Road, Chester, CH1 4BJ, UK – sequence: 3 givenname: Eugene E. surname: Tyrtyshnikov fullname: Tyrtyshnikov, Eugene E. email: eugene.tyrtyshnikov@gmail.com organization: Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina 8, Moscow, 119333, Russia |
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| CitedBy_id | crossref_primary_10_1016_j_amc_2015_09_042 crossref_primary_10_1016_j_cpc_2019_106869 crossref_primary_10_1007_s11075_014_9913_1 crossref_primary_10_1137_130931734 |
| Cites_doi | 10.1007/s00607-009-0047-6 10.1007/s00365-011-9131-1 10.1098/rsta.1998.0248 10.1007/s00041-012-9227-4 10.1016/S0024-3795(01)00336-6 10.1016/j.tcs.2004.01.005 10.1016/0096-3003(95)00270-7 10.1023/A:1016601312158 10.2478/cmam-2011-0021 10.1115/1.3167615 10.1137/0703035 10.1007/s10237-005-0011-0 10.1023/B:NUMA.0000027736.85078.be 10.1137/090752286 10.1109/TAC.1984.1103499 10.1109/ICASSP.1980.1171074 10.1007/s00211-008-0143-0 10.1017/CBO9780511806834 10.1016/j.laa.2011.11.008 10.1002/zamm.19330130111 10.1093/imanum/15.3.357 10.1137/0912058 10.1134/S1064562409040115 10.1137/090757861 10.1007/978-3-7643-8539-2_14 10.1137/0213019 10.1016/j.cma.2004.06.006 10.1007/BF02820620 10.1090/S0025-5718-1965-0178586-1 10.1137/110844830 |
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| Keywords | Tensor train format Superfast Fourier transform 45E10 26A33 Fast convolution Triangular Toeplitz matrix 15A69 65F05 Fractional calculus Divide and conquer |
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| SubjectTerms | Algorithms Approximation Computation Divide and conquer Fast convolution Format Fractional calculus Inversions Mathematical analysis Mathematical models Matrices (mathematics) Superfast Fourier transform Tensor train format Triangular Toeplitz matrix |
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| Title | Superfast solution of linear convolutional Volterra equations using QTT approximation |
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