MiS Preprint Repository

The Kronecker tensor-product approximation combined with the $\mathcal{H} $-matrix techniques provides an efficient tool to represent integral operators as well as a discrete elliptic operator inverse $A^{-1}\in\mathbb{R}^{N\times N}$ in $\mathbb{R}^{d}$ (the discrete Green's function) with a high spatial dimension $d$. In the present paper we give a survey on modern methods of the structured tensor-product approximation to multi-dimensional integral operators and Green's functions and present some new results on the existence of low tensor-rank decompositions to a class of function-related operators. The asymptotic complexity of the considered data-sparse representations is estimated by $\mathcal{O}(d n\log^{q}n)$ with $q$ independent of $d$, where $n=N^{1/d}$ is the dimension of the discrete problem in one space direction. In particular, we apply the results to the Newton, Yukawa and Helmholtz kernels $\frac{1}{|x-y|} $, $\frac{e^{-\lambda|x-y| }}{|x-y|} $ and $\frac{\cos(\lambda|x-y| )}{|x-y|}$, respectively, with $x,y \in\mathbb{R}^{d}$.