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Register a new userIntel Cilk Plus for Complex Parallel Algorithms: Enormous Fast Fourier Transform (EFFT) Library
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Added by
Andrey Vladimirov
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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In this paper we demonstrate the methodology for parallelizing the computation of large one-dimensional discrete fast Fourier transforms (DFFTs) on multi-core Intel Xeon processors. DFFTs based on the recursive Cooley-Tukey method have to control cache utilization, memory bandwidth and vector hardware usage, and at the same time scale across multiple threads or compute nodes. Our method builds on single-threaded Intel Math Kernel Library (MKL) implementation of DFFT, and uses the Intel Cilk Plus framework for thread parallelism. We demonstrate the ability of Intel Cilk Plus to handle parallel recursion with nested loop-centric parallelism without tuning the code to the number of cores or cache metrics. The result of our work is a library called EFFT that performs 1D DFTs of size 2^N for N>=21 faster than the corresponding Intel MKL parallel DFT implementation by up to 1.5x, and faster than FFTW by up to 2.5x. The code of EFFT is available for free download under the GPLv3 license. This work provides a new efficient DFFT implementation, and at the same time demonstrates an educational example of how computer science problems with complex parallel patterns can be optimized for high performance using the Intel Cilk Plus framework.
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In this work, we present two parallel algorithms for the large-scale discrete Fourier transform (DFT) on Tensor Processing Unit (TPU) clusters. The two parallel algorithms are associated with two formulations of DFT: one is based on the Kronecker product, to be specific, dense matrix multiplications between the input data and the Vandermonde matrix, denoted as KDFT in this work; the other is based on the famous Cooley-Tukey algorithm and phase adjustment, denoted as FFT in this work. Both KDFT and FFT formulations take full advantage of TPUs strength in matrix multiplications. The KDFT formulation allows direct use of nonuniform inputs without additional step. In the two parallel algorithms, the same strategy of data decomposition is applied to the input data. Through the data decomposition, the dense matrix multiplications in KDFT and FFT are kept local within TPU cores, which can be performed completely in parallel. The communication among TPU cores is achieved through the one-shuffle scheme in both parallel algorithms, with which sending and receiving data takes place simultaneously between two neighboring cores and along the same direction on the interconnect network. The one-shuffle scheme is designed for the interconnect topology of TPU clusters, minimizing the time required by the communication among TPU cores. Both KDFT and FFT are implemented in TensorFlow. The three-dimensional complex DFT is performed on an example of dimension $8192 times 8192 times 8192$ with a full TPU Pod: the run time of KDFT is 12.66 seconds and that of FFT is 8.3 seconds. Scaling analysis is provided to demonstrate the high parallel efficiency of the two DFT implementations on TPUs.
Study of general purpose computation by GPU (Graphics Processing Unit) can improve the image processing capability of micro-computer system. This paper studies the parallelism of the different stages of decimation in time radix 2 FFT algorithm, designs the butterfly and scramble kernels and implements 2D FFT on GPU. The experiment result demonstrates the validity and advantage over general CPU, especially in the condition of large input size. The approach can also be generalized to other transforms alike.
We propose an all-digital telescope for 21 cm tomography, which combines key advantages of both single dishes and interferometers. The electric field is digitized by antennas on a rectangular grid, after which a series of Fast Fourier Transforms recovers simultaneous multifrequency images of up to half the sky. Thanks to Moores law, the bandwidth up to which this is feasible has now reached about 1 GHz, and will likely continue doubling every couple of years. The main advantages over a single dish telescope are cost and orders of magnitude larger field-of-view, translating into dramatically better sensitivity for large-area surveys. The key advantages over traditional interferometers are cost (the correlator computational cost for an N-element array scales as N log N rather than N^2) and a compact synthesized beam. We argue that 21 cm tomography could be an ideal first application of a very large Fast Fourier Transform Telescope, which would provide both massive sensitivity improvements per dollar and mitigate the off-beam point source foreground problem with its clean beam. Another potentially interesting application is cosmic microwave background polarization.
We introduce PyParSVDfootnote{https://github.com/Romit-Maulik/PyParSVD}, a Python library that implements a streaming, distributed and randomized algorithm for the singular value decomposition. To demonstrate its effectiveness, we extract coherent structures from scientific data. Futhermore, we show weak scaling assessments on up to 256 nodes of the Theta machine at Argonne Leadership Computing Facility, demonstrating potential for large-scale data analyses of practical data sets.
In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with texttt{PDSTEDC} in ScaLAPACK, PSDC is always faster and achieves $1.4$x--$1.6$x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.
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