Numerically Error-Controlled Computation of Gaussian-Beam Propagating Plane-Wave Fast Multipole Translation Operators

The Gaussian beam concept allows to produce well focused propagating plane-wave fast multipole translation operators, where many of the plane-wave samples with small magnitudes can be dropped during the computation of the fast multipole spectral integrals over the Ewald sphere. However, the parameterization of the Gaussian beam operators and of the related numerical evaluation procedure is not easy. In order to overcome the related issues, a fully numerical procedure for the optimization of the Gaussian-beam parameters with appropriate error control is presented. This procedure is numerically very efficient and can be utilized in the pre-computation of fast multipole translation operators within fast multipole or multilevel fast multipole solutions of integral equations. The versatility and accuracy of this approach is demonstrated within a multilevel fast multipole accelerated inverse source-solver, which can efficiently handle large scale source distributions.