Abstract
The technology progress today makes it possible to treat most of the problems of physical geodesy by means of numerical arrangements hardly imaginable earlier. Nevertheless, considering an evaluation of spheroidal (spherical and ellipsoidal) harmonic functions in our typical tasks, we still observe a huge performance gap between our demands and capabilities of common CPUs. Methods used for calculating associated Legendre functions are mostly recursive and thus sequential. Therefore, it is challenging, but feasible, to arrange the processing of Legendre functions in a way that reduces memory utilisation and admits massive parallelism. Following this aim, we developed a streaming-parallel algorithm for computing oblate spheroidal harmonic functions and their derivatives. The algorithm is free of assumptions concerning the function arguments, maximal degree/order or number of computation points and can be utilised on any data type, like a vector or scalar float, double or even integer numbers. Besides, it solves floating-point issues in the numerical treatment of Legendre functions. We demonstrate its Open Computing Language (OpenCL) implementation on a general-purpose graphics processing unit (GPGPU), which is ideal for its inexpensive computational power of some TFlops. Added performance benchmarks lead to the conclusion that our implementation on a single GPGPU device substantially outperforms recent multi-core CPUs, free of any precision penalty. Furthermore, thanks to the OpenCL standard, we can benefit from an excellent portability and scalability over heterogeneous parallel platforms. Let us note finally, that the topic presented is a matter of importance in many other application fields, not only in physical geodesy.
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Notes
- 1.
In this study we are considering GRS 80 (or WGS 84) parameters E = 521 854. 0097 m, a = 6378137 m, \(b^{2} = a^{2} - E^{2}\) as in Moritz (1984).
- 2.
Usually we suppose \(H_{n,m} = \sqrt{(2 -\delta _{0,m } )(2n + 1)\frac{(n-m)!} {(n+m)!}}\).
- 3.
For instance j max < 100 for ρ 2 e 2 < 0. 007 and n < 22, 000.
- 4.
Typical cost of a global memory access is equal to the cost of 200–600 Flops (Floating-point operations) on our GPGPU device, cf. Advanced Micro Devices (2011).
- 5.
For instance, double precision FP numbers with χ = 256 provide the “operational band” 2±256 ≈ 10±77, still allowing to track the numbers as small as \(2^{-549755814000} \approx 10^{-165492990300}\) (for z = 32).
- 6.
Array addressing becomes clear from the source code. Note that the index formula at lines 14 and 25 is used just for a better code readability; the production version of the kernel utilises memory pointers.
- 7.
An inactive workgroup (all the work items in the workgroup do not issue instructions) releases allocated compute resources, enabling an immediate execution of another job waiting in a device queue.
- 8.
The cspqnm_block kernel occupancy factor reaches 69% on our GPGPU, constrained mainly with the local memory size per CU.
- 9.
As Eq. (27) should be applied on \(C_{n,m} \in \mathbb{C}\) in the spheroidal harmonics applications, we substitute it by two separate S m calculations.
- 10.
Run-time criterion ζ N, m ≠ 0 (or resulting ζ m, m ≠ 0 in Clenshaw summation) based on the extended exponent approach is used for a decision about the maximal m relevant in particular synthesis. A little slowdown in enhanced LFRF caused by the exponent extension processing completely diminish with the higher orders cancellation.
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Acknowledgements
The work on this paper was partly supported by the European Regional Development Fund (ERDF), project “NTIS – New Technologies for Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090 and also by the Czech Science Foundation through Project No. 14-34595S. All this support is gratefully acknowledged.
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Nesvadba, O., Holota, P. (2015). An OpenCL Implementation of Ellipsoidal Harmonics. In: Sneeuw, N., Novák, P., Crespi, M., Sansò, F. (eds) VIII Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, vol 142. Springer, Cham. https://doi.org/10.1007/1345_2015_59
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DOI: https://doi.org/10.1007/1345_2015_59
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