Accurate cubature of volume potentials over high-dimensional half-spaces

We study high order cubature formulas for the computation of harmonic potentials over the n-dimensional half-space within the framework of approximate approximations. The cubature of the potentials is reduced to the quadrature of one-dimensional integrals over the half-line. In addition to a formula for general densities with bounded support on the half-space, we derive a tensor product representation of the integral operator which admits efficient cubature procedures for densities with separated approximation in very high space dimensions. Numerical experiments for the half-space up to dimension n = 106 confirm the predicted approximation errors. Bibliography: 11 titles. © 2011 Springer Science+Business Media, Inc.