SIAM Journal on Matrix Analysis and Applications
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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 multidimensional 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 memory space of the considered data-sparse representations is estimated by $\mathcal{O}(dn\log^{q}n)$ with $q$ independent of $d$, retaining the approximation accuracy of order $O(n^{-\delta})$, 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 $x,y \in \mathbb{R}^{d}$.