Regular Article: Computing Fourier Transforms and Convolutions on the 2-Sphere
Advances in Applied Mathematics
A fast transform for spherical harmonics
A fast transform for spherical harmonics
Optimization of computations in global geopotential field applications
ICCS'03 Proceedings of the 2003 international conference on Computational science: PartII
Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization
ICCS '08 Proceedings of the 8th international conference on Computational Science, Part II
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On the sphere, global Fourier transforms are non Abelian and usually called Spherical Harmonic Transforms (SHTs). Discrete SHTs are defined for various grids of data but most applications have requirements in terms of preferred grids and polar considerations. Chebychev quadrature has proven most appropriate in discrete analysis and synthesis to very high degrees and orders. Multiresolution analysis and synthesis that involve convolutions, dilations and decimations are efficiently carried out using SHTs. The high-resolution global datasets becoming available from satellite systems require very high degree and order SHTs for proper representation of the fields. The implied computational efforts in terms of efficiency and reliability are very challenging. The efforts made to compute SHTs and their inverses to degrees and orders 3600 and higher are discussed with special emphasis on numerical stability and information preservation. Parallel and grid computations are imperative for a number of geodetic, geophysical and related applications where near kilometre resolution is required. Parallel computations have been investigated and preliminary results confirm the expectations in terms of efficiency. Further work is continuing on optimizing the computations.