Fast cubature of volume potentials over rectangular domains
It provides a practical method for high-dimensional potential computation, but is incremental as it extends existing approximate approximation techniques to a specific class of potentials.
This paper develops high-order cubature formulas for advection-diffusion potentials over boxes, achieving O(h^6) approximation accuracy in dimensions up to 10^8 by reducing the problem to one-dimensional quadrature via approximate approximations.
In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $O(h^6)$ up to dimension $10^8$.