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
Bibtex
MSC-Numbers: 65F50, 65F30, 46B28, 47A80
Keywords and phrases: boltzmann equation, hierarchical matrices, kronecker tensor product, sinc interpolation
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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 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 - in the one-dimensional problem size n (depending on the model kernel function), which noticeably improves the complexity of the full matrix representation.