The Numerical Evaluation of Multiple Integrals on Parallel Computers

Abstract

Many problems in the applied sciences require the numerical calculation of integrals. Sometimes a problem requires the calculation of a single integral in one or more dimensions, but more often the calculation of a large number of integrals is required, with the integration region fixed for all of the integrals. Two examples of this second class of integration problem occur a) in quantum chemistry when the Self Consistent Field method [14] is used for the study of molecules and large numbers of similar two and three dimensional electron orbital integrals are required, and b) with the finite element method for the numerical solution of partial differential equations [1], where the Galerkin formulation of the method requires the computation of large numbers of one, two or three dimensional integrals before an approximate solution to the partial differential equation can be found. For many problems, the numerical integration occupies a significant proportion of the total processing time. All of the commonly used integration methods (two good general references are [2] and [16]) use a linear combination of the integrand function values to estimate the integrals, so this numerical integration time is usually dominated by the repeated calculation of the integrand values. However, some adaptive algorithms also spend a significant proportion of the computation time with further processing of the integrand values in order to obtain a good subdivision of the integration region.