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

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 S i n c 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 β ⁡ n ) O(n^4log ^β n) - O ( n 5 log β ⁡ n ) O(n^5log ^β n) in the one-dimensional problem size n n (depending on the model kernel function), which noticeably improves the complexity O ( n 6 log β ⁡ n ) O(n^6log ^β n) of the full matrix representation.