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Appl."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication-hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient (CG) method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare it to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.<\/jats:p>","DOI":"10.1137\/17m1117872","type":"journal-article","created":{"date-parts":[[2018,3,13]],"date-time":"2018-03-13T11:12:00Z","timestamp":1520939520000},"page":"426-450","source":"Crossref","is-referenced-by-count":22,"title":["Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method"],"prefix":"10.1137","volume":"39","author":[{"given":"Siegfried","family":"Cools","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emrullah Fatih","family":"Yetkin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emmanuel","family":"Agullo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luc","family":"Giraud","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wim","family":"Vanroose","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,3,13]]},"reference":[{"key":"atypb1","unstructured":"S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, K. Rupp, B. Smith, S. Zampini, and H. Zhang,\n                      PETSc Web page\n                      ,http:\/\/www.mcs.anl.gov\/petsc, 2015."},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst,\n                      Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods\n                      , 2nd ed., SIAM, Philadelphia, PA, 1994.","DOI":"10.1137\/1.9781611971538"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/120893057"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/120881191"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90045-9"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/nla.643"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8191(96)00022-1"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"E. D'Azevedo, V. Eijkhout, and C. Romine,\n                      Reducing Communication Costs in the Conjugate Gradient Algorithm on Distributed Memory Multiprocessors\n                      , Technical report TM\/12192, Oak Ridge National Lab, Oak Ridge, TN, 1992.","DOI":"10.2172\/7172467"},{"key":"atypb10","unstructured":"E. de Sturler,\n                      A parallel variant of GMRES(m)\n                      , in Proceedings of the 13th IMACS World Congress on Computational and Applied Mathematics, Vol. 9, Dublin, Ireland, Criterion Press, 1991, pp. 682-683."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(95)00079-A"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"J. Demmel,\n                      Applied Numerical Linear Algebra\n                      , SIAM, Philadelphia, PA, 1997.","DOI":"10.1137\/1.9781611971446"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1017\/S096249290000235X"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1177\/1094342010391989"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"J. Dongarra, I. Duff, D. Sorensen, and H. Van der Vorst,\n                      Numerical Linear Algebra for High-Performance Computers\n                      , SIAM, Philadelphia, PA, 1998.","DOI":"10.1137\/1.9780898719611"},{"key":"atypb16","unstructured":"J. Dongarra and M. Heroux,\n                      Toward a New Metric for Ranking High Performance Computing Systems\n                      , Technical report SAND2013-4744, 312, Sandia National Laboratories, Livermore, CA, 2013."},{"key":"atypb17","unstructured":"J. Dongarra, M. Heroux, and P. Luszczek,\n                      HPCG Benchmark: A New Metric for Ranking High Performance Computing Systems\n                      , Knoxville, technical report UT-EECS-15-736, University of Tennessee, 2015."},{"key":"atypb18","first-page":"204","author":"Eller P.","year":"2016","journal-title":"Storage and Analysis, IEEE"},{"key":"atypb19","first-page":"160","volume":"3","author":"Erhel J.","year":"1995","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-013-9713-z"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/12086563X"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.parco.2013.06.001"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90285-1"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895284944"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"A. Greenbaum,\n                      Iterative Methods for Solving Linear Systems\n                      , SIAM, Philadelphia, PA, 1997.","DOI":"10.1137\/1.9781611970937"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/0613011"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897331862"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"N. Higham,\n                      Accuracy and Stability of Numerical Algorithms\n                      , SIAM, Philadelphia, PA, 2002.","DOI":"10.1137\/1.9780898718027"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"J. Liesen and Z. Strako\u0161,\n                      Krylov Subspace Methods: Principles and Analysis\n                      , Oxford University Press, Oxford, 2012.","DOI":"10.1093\/acprof:oso\/9780199655410.001.0001"},{"key":"atypb32","unstructured":"G. Meurant,\n                      Computer Solution of Large Linear Systems\n                      , Vol. 28, Elsevier, Amsterdam, 1999."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1017\/S096249290626001X"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385754"},{"key":"atypb35","unstructured":"C. Paige,\n                      The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices\n                      , Ph.D. thesis, University of London, London, 1971."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/10.3.373"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/18.3.341"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90167-6"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , SIAM, Philadelphia, PA, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/BF02309342"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/BF02141261"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897323087"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/BF02564277"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8191(87)90051-2"},{"key":"atypb45","first-page":"56","volume":"13","author":"Strako\u0161 Z.","year":"2002","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-005-0032-1"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01171-0"},{"key":"atypb48","doi-asserted-by":"crossref","unstructured":"H. Van der Vorst,\n                      Iterative Krylov Methods for Large Linear Systems\n                      , Vol. 13, Cambridge University Press, Cambridge, 2003.","DOI":"10.1017\/CBO9780511615115"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599353865"},{"key":"atypb50","unstructured":"J. Wilkinson,\n                      Rounding Errors in Algebraic Processes\n                      , Courier Corporation, North Chelms ford, MA, 1994."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1117872","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:09:42Z","timestamp":1787324982000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1117872"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":49,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1117872"],"URL":"https:\/\/doi.org\/10.1137\/17m1117872","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}