MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Boris N. Khoromskij


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 $O(n^4\log^\beta n)$ - $O(n^5\log^\beta n)$ 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.

MSC Codes:
65F50, 65F30, 46B28, 47A80
boltzmann equation, hierarchical matrices, kronecker tensor product, sinc interpolation

Related publications

2007 Repository Open Access
Boris N. Khoromskij

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

In: Mathematics of computation, 76 (2007) 259, pp. 1291-1315