Sphere quadrature for orientation averaging

[GoogleCodeExporter]

Implement orientation averaging using quadrature over sphere surface,
according to ideas of Antti Penttila.

Original issue reported on code.google.com by yurkin on 24 Dec 2008 at 7:16

[myurkin]

/cc @anatheana

[myurkin]

Another relevant discussion is https://groups.google.com/forum/#!topic/adda-discuss/9gJTy2spRjo

[myurkin]

Another idea is to use Fibonacci lattice (kind of quasi Monte Carlo) - see
Zhang J. On numerical orientation averaging with spherical Fibonacci point sets and compressive scheme, J. Quant. Spectrosc. Radiat. Transfer 206, 1–7 (2018).

This paper also applies compressed sensing, based on the expansion of orientation-dependent scattered quantity into series of spherical harmonics. In this respect, this is related to #138

[myurkin]

Some general ideas how to evaluate different quadratures (or more specifically, cubatures) for the DDA and other light-scattering simulation methods:
Yurkin M.A. Fair evaluation of orientation-averaging techniques in light-scattering simulations: Comment on “Evaluation of higher-order quadrature schemes in improving computational efficiency for orientation-averaged single-scattering properties of nonspherical ice particles” by Fenni et al., J. Geophys. Res. Atmos. 128, e2021JD036088 (2023).

[myurkin]

Another recent relevant paper is
Fazel-Najafabadi A. and Auguié B. Orientation-averaged light scattering by nanoparticle clusters: Far-field and near-field benchmarks of numerical cubature methods, J. Quant. Spectrosc. Radiat. Transfer 286, 108197 (2022).
It benchmarks several cubatures and also mentions (with references) several adaptive options. The latter may be an ultimate goal for solving this issue.

[myurkin]

This paper has a concise description of various cubatures, including their efficiency (number of point to integrate spherical harmonics up to a given point): https://cbeentjes.github.io/files/Ramblings/QuadratureSphere.pdf