Preprint 4/2005

Structured Data-Sparse Approximation to High Order Tensors Arising from the Deterministic Boltzmann Equation

Boris N. Khoromskij

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Submission date: 10. Jan. 2005 (revised version: February 2005)
Pages: 31
published in: Mathematics of computation, 76 (2007) 259, p. 1291-1315 
DOI number (of the published article): 10.1090/S0025-5718-07-01901-1
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: boltzmann equation, hierarchical matrices, kronecker tensor product, sinc interpolation
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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 multi-variate 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 formula5 - formula7 in the one-dimensional problem size n (depending on the model kernel function), which noticeably improves the complexity formula11 of the full matrix representation.

20.02.2013, 14:49