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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.