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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation
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by Boris N. Khoromskij PDF
Math. Comp. 76 (2007), 1291-1315 Request permission

Abstract:

We develop efficient data-sparse representations to a class of high order tensors via a block many-fold Kronecker product decomposition. Such a decomposition is based on low separation-rank approximations of the corresponding multivariate generating function. We combine the $Sinc$ interpolation and a quadrature-based approximation with hierarchically organised block tensor-product formats. Different matrix and tensor operations in the generalised Kronecker tensor-product format including the Hadamard-type product can be implemented with the low cost. An application to the collision integral from the deterministic Boltzmann equation leads to an asymptotical cost $O(n^4\log ^\beta n)$ - $O(n^5\log ^\beta n)$ in the one-dimensional problem size $n$ (depending on the model kernel function), which noticeably improves the complexity $O(n^6\log ^\beta n)$ of the full matrix representation.
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Additional Information
  • Boris N. Khoromskij
  • Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
  • Email: bokh@mis.mpg.de
  • Received by editor(s): February 22, 2005
  • Received by editor(s) in revised form: October 4, 2005
  • Published electronically: February 16, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Math. Comp. 76 (2007), 1291-1315
  • MSC (2000): Primary 65F50, 65F30; Secondary 15A24, 15A99
  • DOI: https://doi.org/10.1090/S0025-5718-07-01901-1
  • MathSciNet review: 2299775