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Optimization of Triangular and Banded Matrix Operations Using 2d-Packed Layouts

Toufik Baroudi, Rachid Seghir, Vincent Loechner
<span title="2017-12-18">2017</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jfrn2kjyarhe7npmgvoxdp4cxu" style="color: black;">ACM Transactions on Architecture and Code Optimization (TACO)</a> </i> &nbsp;
Additional Key Words and Phrases: Polyhedral Model, Code optimization and parallelization, Sparse matrices, 2d-packed layouts.  ...  This is mainly due to the shortage of optimization techniques dealing with irregular control structures.  ...  On one hand, because they have to write the banded and triangular matrix computations in packed format (see Figure 2b for example), which they may not be familiar with.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3162016">doi:10.1145/3162016</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oh5pix2t7baf7p7gebp3gw2gue">fatcat:oh5pix2t7baf7p7gebp3gw2gue</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180726035137/https://hal.inria.fr/hal-01633724/file/BSL17-2dpacked.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/16/a5/16a58776841cc72b366032f8b287a0e6cb7aab08.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3162016"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

An Algorithm-by-Blocks for SuperMatrix Band Cholesky Factorization [chapter]

Gregorio Quintana-Ortí, Enrique S. Quintana-Ortí, Alfredo Remón, Robert A. van de Geijn
<span title="">2008</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Experimental results for the Cholesky factorization of band matrices on two parallel platforms with sixteen processors demonstrate the scalability of the solution.  ...  We pursue the scalable parallel implementation of the factorization of band matrices with medium to large bandwidth targeting SMP and multi-core architectures.  ...  This work was partially carried out when Alfredo Remn was visiting the Chemnitz University of Technology with a grant from the programme Plan 2007 de Promocin de la Investigacin of the Universidad Jaime  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-92859-1_21">doi:10.1007/978-3-540-92859-1_21</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4jv4sopocncjpm3a665pn3jjh4">fatcat:4jv4sopocncjpm3a665pn3jjh4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180720121842/http://www.cs.utexas.edu/users/flame/pubs/VECPAR2008.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b7/cc/b7ccb33d4ba8cab6ae98d5b4a3ea8e99d7c3e430.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-92859-1_21"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Parallel computation of a Krylov matrix for a sparse and structured input

V.Y. Pan
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/q5bfnofiffaoldjsebw7gavjjm" style="color: black;">Mathematical and computer modelling</a> </i> &nbsp;
As in our previous work, we reduce parallel computation of a Krylov matrix to solving a parametrized linear system of equations.  ...  This time we show that such a method is effective in the cases of banded matrices, sparse and structured matrices and triangular matrices.  ...  To evaluate the matrix K(A, v, m) on a parallel computer (assuming m = 2h -1 for simplicity), one may recursively compute the matrices A2", i = 1,. . . , h -1, and then A2(v, Au) = (A2v, A3v) , A4 (v,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0895-7177(95)00084-f">doi:10.1016/0895-7177(95)00084-f</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2rv2mevjanbnrkqxwtyigwrnlm">fatcat:2rv2mevjanbnrkqxwtyigwrnlm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20040620214836/http://comet.lehman.cuny.edu:80/vpan/pdf/pan123.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e7/59/e759db6e1016c46dd37477dcc84821649ac437d5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0895-7177(95)00084-f"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Parallel Sub-structuring Methods for Solving Sparse Linear Systems on a Cluster of GPUs

Abal- Kassim Cheik Ahamed, Frederic Magoules
<span title="">2014</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/huq76n6umbcx3beqwcclmdlika" style="color: black;">2014 IEEE Intl Conf on High Performance Computing and Communications, 2014 IEEE 6th Intl Symp on Cyberspace Safety and Security, 2014 IEEE 11th Intl Conf on Embedded Software and Syst (HPCC,CSS,ICESS)</a> </i> &nbsp;
GPU computing, with CUDA, is used to accelerate the operations performed on each processor. Numerical experiments have been performed on a set of matrices arising from engineering problems.  ...  The main objective of this work consists in analyzing sub-structuring method for the parallel solution of sparse linear systems with matrices arising from the discretization of partial differential equations  ...  Acknowledgment The authors acknowledge the CUDA Research Center at Ecole Centrale Paris (France) for its support and for providing the computing facilities.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/hpcc.2014.24">doi:10.1109/hpcc.2014.24</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/hpcc/AhamedM14a.html">dblp:conf/hpcc/AhamedM14a</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vqp5xwaxpbhcjng4crc4q32ami">fatcat:vqp5xwaxpbhcjng4crc4q32ami</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210901101713/https://arxiv.org/pdf/2108.13162v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c7/bc/c7bc2febf65a9355d57db0496f2a3a5b60433894.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/hpcc.2014.24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Page 1159 of Mathematical Reviews Vol. , Issue 90B [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(I-BERG) Numerical experiments with ABS algorithms for linear systems on a parallel machine. J. Optim. Theory Appl. 60 (1989), no. 3, 375-392.  ...  Summary: “Numerical results are obtained on sequential and par- allel versions of ABS algorithms for linear systems for both full matrices and g-band matrices.  ... 
<span class="external-identifiers"> </span>
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Page 383 of Mathematical Reviews Vol. , Issue 91A [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(BG-AOS) On some ways of approximating inverses of banded matrices in connection with deriving preconditioners based on incomplete block factorizations.  ...  In fact, for systems with m equations and n unknowns, each iteration may be computed in parallel time O(log m + log n), on an EREW PRAM with O(mn) processors.” 91a:65076 65F10 93B40 Baumeister, J.  ... 
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Enhancing Parallelism of Tile Bidiagonal Transformation on Multicore Architectures Using Tree Reduction [chapter]

Hatem Ltaief, Piotr Luszczek, Jack Dongarra
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
First introduced by Ltaief et. al [23] , the bidiagonal transformation using tile algorithms with a two-stage approach has shown very promising results on square matrices.  ...  The objective of this paper is to enhance the parallelism of the tile bidiagonal transformation using tree reduction on multicore architectures.  ...  Each subdomain computation proceeds locally in parallel.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-31464-3_67">doi:10.1007/978-3-642-31464-3_67</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r4mvjszqpngytjf7527sbyrhzm">fatcat:r4mvjszqpngytjf7527sbyrhzm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170811110414/http://www.netlib.org/utk/people/JackDongarra-20130-07-11/PAPERS/ppam2011.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/96/e0/96e052147534e27e5d75d448af0504b597d6544c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-31464-3_67"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Efficient Parallel Shortest Path Algorithms for Banded Matrices

Y. Han, Y. Igarashi
<span title="">1993</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3qrmigawsbhbljazwy2rerl5ni" style="color: black;">1993 International Conference on Parallel Processing - ICPP&#39;93 Vol3</a> </i> &nbsp;
We present efficient parallel shortest path algorithms for an $n\cross n$  ...  A common approach used in parallel computation is to put these matrices at the leaves of a binary tree and the matrix multiplications proceed as dictated by the tree.  ...  This approach does not work for banded matrices because when matrices are multiplied together we are essentially filling the entries outside the band of the matrix.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icpp.1993.73">doi:10.1109/icpp.1993.73</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icpp/HanI93.html">dblp:conf/icpp/HanI93</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m6im2ymvv5ek3jruyae4jsxs2a">fatcat:m6im2ymvv5ek3jruyae4jsxs2a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922072844/https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/84041/1/0871-25.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/dc/5c/dc5c32140b0dbb0d52e5adca09eacfa1baf45dc3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/icpp.1993.73"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Tight bounds on one- and two-pass MapReduce algorithms for matrix multiplication

Prakash Ramanan, Ashita Nagar
<span title="">2016</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vxrc3vebzzachiwy3nopwi3h5u" style="color: black;">Proceedings of the 3rd ACM SIGMOD Workshop on Algorithms and Systems for MapReduce and Beyond - BeyondMR &#39;16</a> </i> &nbsp;
CCS Concepts •Theory of computation → MapReduce algorithms; Keywords Matrix multiplication; lower bounds Reducer Size q [1]: This is one measure of the degree of parallelism.  ...  It measures communication cost using the replication rate r, and measures parallelism by reducer size q. It gives a tight bound on qr for multiplying dense square matrices.  ...  We can extend Theorem 7.1 to rectangular matrices, with very little modification. We want to compute Cm×p = Am×nBn×p.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2926534.2926542">doi:10.1145/2926534.2926542</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/sigmod/RamananN16.html">dblp:conf/sigmod/RamananN16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ldqvjwozbncetj7le2snw2bhqy">fatcat:ldqvjwozbncetj7le2snw2bhqy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190218131320/https://static.aminer.org/pdf/20170130/pdfs/sigmod/rfavbxojkiiq6nucqd8d2e7vz0trss3y.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/55/65/5565f7e44198b79200d200de154e2557ef16ccb5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/2926534.2926542"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

The Chunks and Tasks Matrix Library 2.0 [article]

Emanuel H. Rubensson, Elias Rudberg, Anastasia Kruchinina, Anton G. Artemov
<span title="2020-11-23">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present a C++ header-only parallel sparse matrix library, based on sparse quadtree representation of matrices using the Chunks and Tasks programming model.  ...  This is demonstrated for the example of block-sparse matrix-matrix multiplication applied to three sequences of matrices with different nonzero structure, using the CHT-MPI 2.0 runtime library implementation  ...  Computational resources were provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High Performance Computing, KTH Royal Institute of Technology in Stockholm.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2011.11762v1">arXiv:2011.11762v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mjrowql7rvcdnmdmk6vagxgdp4">fatcat:mjrowql7rvcdnmdmk6vagxgdp4</a> </span>
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A Bit-Compatible Shared Memory Parallelization for ILU(k) Preconditioning and a Bit-Compatible Generalization to Distributed Memory [article]

Xin Dong, Gene Cooperman
<span title="2011-05-12">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In experiments on a 16-core computer, the enhanced TPILU(k)-based iterative linear solver performed up to 9 times faster.  ...  For example, the new algorithm achieves 50 times speedup with 80 nodes for general sparse matrices of dimension 160,000 that are diagonally dominant.  ...  From matrices L and U, one efficiently computes A −1 as U −1 L −1 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0803.0048v4">arXiv:0803.0048v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bjwdfsqucfbrjb3aiwgeecfo4e">fatcat:bjwdfsqucfbrjb3aiwgeecfo4e</a> </span>
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Time-Memory Trade-Offs Using Sparse Matrix Methods for Large-Scale Eigenvalue Problems [chapter]

Keita Teranishi, Padma Raghavan, Chao Yang
<span title="">2003</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Our method also allows a more scalable parallel formulation for eigenvector computation through spectrum slicing.  ...  We show that by using sparse matrix methods, a direct-iterative hybrid scheme can significantly reduce memory requirements while requiring less computational time than a banded direct scheme.  ...  Acknowledgments We greatfully acknowledge several useful discussions with J. L. Barlow at the Pennsylvania State University and R. C. Ward and W. Gansterer at the University of Tennessee.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-44839-x_88">doi:10.1007/3-540-44839-x_88</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yus2i46h2ng7jfo4nbrx7g2ds4">fatcat:yus2i46h2ng7jfo4nbrx7g2ds4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20090116025028/http://www.cse.psu.edu/~teranish/paper/iccsa2003.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0a/12/0a12d2b5f165a924cd172cf3fc8bc41b3ebfc718.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-44839-x_88"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A communication-avoiding parallel algorithm for the symmetric eigenvalue problem [article]

Edgar Solomonik, Grey Ballard, James Demmel, Torsten Hoefler
<span title="2016-04-13">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The algorithm first reduces the dense symmetric matrix to a banded matrix with the same eigenvalues. Subsequently, the algorithm employs successive reduction to O( p) thinner banded matrices.  ...  We employ two new parallel algorithms that achieve lower communication costs for the full-to-band and band-to-band reductions.  ...  Our algorithms focus on reducing the symmetric matrix to thinner and thinner banded matrices with the same eigenvalues.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1604.03703v1">arXiv:1604.03703v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/53gz2x2x4fbyhogw6tvu563aty">fatcat:53gz2x2x4fbyhogw6tvu563aty</a> </span>
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The scheduling of sparse matrix-vector multiplication on a massively parallel DAP computer

J Andersen, G Mitra, D Parkinson
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/sv4mpg7lmfaqdp24ohp5qqiobm" style="color: black;">Parallel Computing</a> </i> &nbsp;
This approach seeks to reduce the inter-processor data movements and organises the operations in batches of massively parallel steps by a heuristic scheduling procedure performed on the host computer.  ...  With these definitions, we can represent the m'th component of y k in terms of the parallel objects:  ...  To deal with such matrices we introduce a new level of aggregation; the block band.  ... 
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Parallel Multicore CSB Format and Its Sparse Matrix Vector Multiplication

Bing Yang, Shuo Gu, Tong-Xiang Gu, Cong Zheng, Xing-Ping Liu
<span title="">2014</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nyihf42pnnaflcw6inocggtm7a" style="color: black;">Advances in Linear Algebra and Matrix Theory</a> </i> &nbsp;
The results show that our parallel multicore CSB format and SpMV algorithm can reach high speedup, and they are highly scalable for banded matrices. sparse matrix-vector multiplication (SpMV) is the most  ...  This paper presents a parallel multicore CSB format and SpMV based on it. We carried out numerical experiments on a parallel multicore computer.  ...  Banded Matrices Stored in Column For the banded matrix, we can optimize the algorithm more efficiently. • Compress the Row Index As shown in Figure 1 , what we need deal with is the area bounded by solid  ... 
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