On the computation of high-dimensional potentials of advection-diffusion operators

We study a fast method for computing potentials of advection-diffusion operators $-D +2bcdot abla+c$ with $binC^n$ and $cin C$ over rectangular boxes in R^n. By combining high-order cubature formulas with modern methods of structured tensor product approximations we derive an approximation of the potentials which is accurate and provides approximation formulas of high-order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands 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 tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in high-dimensional case.