Accurate computation of the high dimensional diffraction potential over hyper-rectangles

Abstract

We propose a fast method for high order approximation of potentials of the Helmholtz type operator ∆ + κ 2 over hyper-rectangles in R n . By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of onedimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and κ 2 = 100.