Approximate approximations: recent developments in the computation of high dimensional potentials

The paper is devoted to a fast method of an arbitrary high order for approximating volume potentials that is successful also in the high dimensional case. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. As basis functions, we choose products of Gaussians and special polynomials, for which the action of integral operators can be written as one-dimensional integrals with a separable integrand, i.e., a product of functions depending only on one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a separable representation of the integral operator. Since only one-dimensional operations are used, the resulting method is efficient also in the high dimensional case. We show how this new approach can be applied to the cubature of polyharmonic potentials, to potentials of elliptic differential operators acting on densities on hyper-rectangular domains, and to parabolic problems.