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Mathematics of Computation

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Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation

Author: Boris N. Khoromskij
Journal: Math. Comp. 76 (2007), 1291-1315
MSC (2000): Primary 65F50, 65F30; Secondary 15A24, 15A99
Published electronically: February 16, 2007
MathSciNet review: 2299775
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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

Keywords: Boltzmann equation, hierarchical matrices, Kronecker tensor product, high order tensors, $sinc$ interpolation and quadratures.
Received by editor(s): February 22, 2005
Received by editor(s) in revised form: October 4, 2005
Published electronically: February 16, 2007
Article copyright: © Copyright 2007 American Mathematical Society