TENSOR-PRODUCT APPROXIMATION TO MULTIDIMENSIONAL INTEGRAL OPERATORS AND GREEN'S FUNCTIONS

The Kronecker tensor-product approximation combined with the H-matrix techniques provides an efficient tool to represent integral operators as well as a discrete elliptic operator inverse A(-1) is an element of R-NxN in 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 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 1/|x-y|, e-lambda|x-y|/|x-y|, and cos(lambda|x-y|)/|x-y|, respectively with x,y is an element of R-d.