Accurate computation of the high dimensional diffraction potential over hyper-rectangles
For computational scientists needing efficient high-dimensional integral operators, this method offers a practical tensor-product approach with demonstrated accuracy.
The paper presents a fast method for high-order approximation of Helmholtz-type potentials over hyper-rectangles in high dimensions, achieving order-6 accuracy up to dimension 100.
We propose a fast method for high order approximation of potentials of the Helmholtz type operator Delta+kappa^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 one-dimensional 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 kappa^2=100.