Parallel Algorithms for the Spectral Transform Method

The spectral transform method is a standard numerical technique for solving partial differential equations on a sphere and is widely used in atmospheric circulation models. Recent research has identified several promising algorithms for implementing this method on massively parallel computers; however, no detailed comparison of the different algorithms has previously been attempted. In this paper, we describe these different parallel algorithms and report on computational experiments that we have conducted to evaluate their efficiency on parallel computers. The experiments used a testbed code that solves the nonlinear shallow water equations on a sphere; considerable care was taken to ensure that the experiments provide a fair comparison of the different algorithms and that the results are relevant to global models. We focus on hypercube- and mesh-connected multicomputers with cut-through routing, such as the Intel iPSC/860, DELTA, and Paragon, and the nCUBE/2, but we also indicate how the results extend to other parallel computer architectures. The results of this study are relevant not only to the spectral transform method but also to multidimensional fast Fourier transforms (FFTs) and other parallel transforms.

[1]  Richard B. Pelz,et al.  A Balanced, Parallel algorithm for Spectral Global Climate Models , 1993, PPSC.

[2]  Thomas H. Dunigan,et al.  Performance of the Intel iPSC/860 and Ncube 6400 hypercubes , 1991, Parallel Comput..

[3]  Ian Foster,et al.  Parallel Spectral Transform Shallow Water Model: a runtime-tunable parallel benchmark code , 1994, Proceedings of IEEE Scalable High Performance Computing Conference.

[4]  J. O. Eklundh,et al.  A Fast Computer Method for Matrix Transposing , 1972, IEEE Transactions on Computers.

[5]  P. Rasch,et al.  Description of the NCAR community climate model (CCM2), June 1993. Technical note , 1993 .

[6]  G. L. Browning,et al.  A comparison of three numerical methods for solving differential equations on the sphere , 1989 .

[7]  William Gropp,et al.  Parallel Scalability of the Spectral Transform Method , 1991, PPSC.

[8]  Harold S. Stone,et al.  Parallel Processing with the Perfect Shuffle , 1971, IEEE Transactions on Computers.

[9]  James J. Hack,et al.  Description of a Global Shallow Water Model Based on the Spectral Transform Method , 1992 .

[10]  G. C. Fox,et al.  Solving Problems on Concurrent Processors , 1988 .

[11]  James J. Hack,et al.  The message passing version of the parallel Community Climate Model , 1994 .

[12]  Anton Schüller,et al.  Parallelizing the ECMWF's Weather Forecast Program: the 2D Case , 1993, Parallel Comput..

[13]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[14]  T. H. Dunigan.  Communication performance of the Intel Touchstone DELTA mesh , 1992 .

[15]  Mohammad Zubair,et al.  A General Purpose Subroutine for Fast Fourier Transform on a Distributed Memory Parallel Machine , 1994, Parallel Comput..

[16]  D. K. Maretis,et al.  Dawn of Massively Parallel Processing in Meteorology , 1990 .

[17]  S. Lennart Johnsson,et al.  Algorithms for Matrix Transposition on Boolean n-Cube Configured Ensemble Architectures , 1988, ICPP.

[18]  Vipin Kumar,et al.  The Scalability of FFT on Parallel Computers , 1993, IEEE Trans. Parallel Distributed Syst..

[19]  W. Washington,et al.  An Introduction to Three-Dimensional Climate Modeling , 1986 .

[20]  Ian T. Foster,et al.  Design and Performance of a Scalable Parallel Community Climate Model , 1995, Parallel Comput..

[21]  Marshall C. Pease,et al.  An Adaptation of the Fast Fourier Transform for Parallel Processing , 1968, JACM.

[22]  Patrick H. Worley,et al.  Parallelizing the spectral transform method. Part II , 1992, Concurr. Pract. Exp..

[23]  R. A. van de Geijn,et al.  Efficient Global Combine Operations , 1991 .

[24]  Paul N. Swarztrauber,et al.  Multiprocessor FFTs , 1987, Parallel Comput..

[25]  S. Takkella,et al.  Complete exchange and broadcast algorithms for meshes , 1994, Proceedings of IEEE Scalable High Performance Computing Conference.

[26]  William L. Briggs,et al.  Bluestein's FFT for arbitrary N on the hypercube , 1991, Parallel Comput..

[27]  James J. Hack,et al.  PCCM2: A GCM adapted for scalable parallel computers , 1994 .

[28]  David H. Bailey,et al.  FFTs in external or hierarchical memory , 1989, Proceedings of the 1989 ACM/IEEE Conference on Supercomputing (Supercomputing '89).

[29]  J. Watts,et al.  Interprocessor collective communication library (InterCom) , 1994, Proceedings of IEEE Scalable High Performance Computing Conference.

[30]  Ian T. Foster,et al.  Parallelizing the Spectral Transform Method: A Comparison of Alternative Parallel Algorithms , 1993, PPSC.

[31]  Alan Edelman,et al.  Optimal Matrix Transposition and Bit Reversal on Hypercubes: All-to-All Personalized Communication , 1991, J. Parallel Distributed Comput..

[32]  R. Bracewell.  The Fourier transform. , 1989, Scientific American.