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14th IEEE International Conference on High Performance Computing and Communications (HPCC) / IEEE 9th International Conference on Embedded Software and Systems (ICESS)

作者： Takahashi, Daisuke Uno, Atsuya Yokokawa, Mitsuo Univ Tsukuba Fac Engn Tsukuba Ibaraki 3058573 Japan Next Generat Supercomp R&D Ctr Dev Grp Kobe Hyogo 6500047 Japan

ISBN: (纸本)9780769547497

In this paper, we propose an implementation of a parallel one-dimensional fast Fourier transform (FFT) on the K computer. The proposed algorithm is based on the six-step FFT algorithm, which can be altered into the recursive six-step FFT algorithm to reduce the number of cache misses. The recursive six-step FFT algorithm improves performance by utilizing the cache memory effectively. We use the recursive six-step FFT algorithm to implement the parallel one-dimensional FFT algorithm. The performance results of one-dimensional FFTs on the K computer are reported. We successfully achieved a performance of over 18 TFlops on 8192 nodes of the K computer (82944 nodes, 128 GFlops/node, 10.6 PFlops peak performance) for a 2(41)-point FFT.

关键词： Fast Fourier transform distributed-memory parallel computer all-to-all communication

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parallel implementation of multiple-precision arithmetic and 2,576,980,370,000 decimal digits of π calculation

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parallel COMPUTING 2010年第8期36卷 439-448页

作者： Takahashi, Daisuke Univ Tsukuba Grad Sch Syst & Informat Engn Tsukuba Ibaraki 3058573 Japan

We present efficient parallel algorithms for multiple-precision arithmetic operations of more than several million decimal digits on distributed-memory parallel computers. A parallel implementation of floating-point real FFT-based multiplication is used, since the key operation for fast multiple-precision arithmetic is multiplication. The operation for releasing propagated carries and borrows in multiple-precision addition, subtraction and multiplication was also parallelized. More than 2.576 trillion decimal digits of pi were computed on 640 nodes of Appro Xtreme-X3 (648 nodes, 147.2 GFlops/node, 95.4 TFlops peak performance) with a computing elapsed time of 73 h 36 min which includes the time required for verification. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Multiple-precision arithmetic Fast Fourier transform distributed-memory parallel computer

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A parallel 1-D FFT algorithm for the Hitachi SR8000

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parallel COMPUTING 2003年第6期29卷 679-690页

作者： Takahashi, D Univ Tsukuba Inst Informat Sci & Elect Tsukuba Ibaraki 3058573 Japan

In this paper, we propose a high-performance parallel one-dimensional fast Fourier transform (FFT) algorithm on clusters of vector symmetric multiprocessor (SMP) nodes. The four-step FFT algorithm can be altered into a five-step FFT algorithm to expand the innermost loop length. We use the five-step algorithm to implement the parallel one-dimensional FFT algorithm. In our proposed parallel FFT algorithm, since we use cyclic distribution, all-to-all communication takes place only once. Moreover, the input data and output data are both in natural order. Performance results of one-dimensional power-of-two FFTs on clusters of pseudo-vector SNIP nodes, Hitachi SR8000, are reported. We succeeded in obtaining performance of over 61 GFLOPS on a 16-node Hitachi SR8000/MPP. (C) 2003 Elsevier Science B.V. All rights reserved.

关键词： fast Fourier transform distributed-memory parallel computer vector symmetric multiprocessor cyclic distribution all-to-all communication

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High-performance radix-2, 3 and 5 parallel 1-D complex FFT algorithms for distributed-memory parallel computers

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JOURNAL OF SUPERCOMPUTING 2000年第2期15卷 207-228页

作者： Takahashi, D Kanada, Y Univ Tokyo Ctr Comp Bunkyo Ku Tokyo 1138658 Japan

In this paper, we propose high-performance radix-2, 3 and 5 parallel 1-D complex FFT algorithms for distributed-memory parallel computers. We use the four-step or six-step FFT algorithms to implement the radix-2, 3 and 5 parallel 1-D complex FFT algorithms. In our parallel FFT algorithms, since we use cyclic distribution, all-to-all communication takes place only once. Moreover, the input data and output data are both in natural order. We also show that the suitability of a parallel FFT algorithm is machine-dependent because of the differences in the architecture of the processor elements in distributed-memory parallel computers. Experimental results of 2(p)3(q)5(r) point FFTs on distributed-memory parallel computers, HITACHI SR2201 and IBM SP2 are reported. We succeeded to get performances of about 130 GFLOPS on a 1024PE HITACHI SR2201 and about 1.25 GFLOPS on a 32PE IBM SP2.

关键词： fast Fourier transform radix-2, 3 and 5 distributed-memory parallel computer cyclic distribution all-to-all communication

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A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1999年第5期20卷 1778-1796页

作者： Rossi, T Toivanen, J Univ Jyvaskyla Dept Math Comp Sci Lab FIN-40351 Jyvaskyla Finland

A parallel fast direct solution method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the five-point finite difference scheme or the piecewise linear finite elements on a triangulated, possibly nonuniform rectangular mesh. The method under consideration has the arithmetical complexity O(N log N), and it is closely related to the cyclic reduction method, but instead of using the matrix polynomial factorization, the so-called partial solution technique is employed. Hence, in this paper, the method is called the partial solution variant of the cyclic reduction method (PSCR method). The method is presented and analyzed in a general radix-q framework and, based on this analysis, the radix-4 variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential efficiency and numerical stability of the PSCR method compared to the well-known BLKTRI implementation of the generalized cyclic reduction method. The good scalability properties of the parallel PSCR method are demonstrated in a distributed-memory Cray T3E-750 computer.

关键词： separable matrix fast Poisson solver cyclic reduction partial solution problem distributed-memory parallel computer parallel computing

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