Comparison and Improvement of the Computational Efficiencies of Two FFT-Based Iterative Solution Methods for the Scalar Multiple-Scattering Equation (Communications in Computational Physics, 2018)
We consider two existing FFT-based fast-convolution iterative solution techniques for the scalar T-matrix multiple-scattering equation [1]. The use of the FFT operation requires field values be expressed on a regular Cartesian grid and the two techniques differ in how to go about achieving this. The first technique [6, 7] uses the non-diagonal translation operator [1, 9] of the spherical multipole field, while the secondmethod [11] uses the diagonal translation operator of Rokhlin [10]. Because of its useof the non-diagonal translator, the first technique has been thought to require a greaternumber of spatial convolutions than the second technique. We establish that the firstmethod requires only half as many convolution operations as the second method for acomparable numerical accuracy and demonstrate, based on an actual CPU time comparison, that it can therefore perform iterations faster than the second method. We thenconsider the respective symmetry relations of the non-diagonal and diagonal translators and discuss a memory-reduction procedure for both FFT-based methods. In thisprocedure, we need to store only the minimum sets of near-field and far-field translation operators and generate missing elements on the fly using the symmetry relations.We show that the relative cost of generating the missing elements becomes smaller asthe number of scatterers increases.