{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:37Z","timestamp":1787337097238,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization.<\/jats:p>","DOI":"10.1137\/050635547","type":"journal-article","created":{"date-parts":[[2006,3,15]],"date-time":"2006-03-15T21:00:21Z","timestamp":1142456421000},"page":"224-240","source":"Crossref","is-referenced-by-count":22,"title":["Topological Optimization of the Evaluation of Finite Element Matrices"],"prefix":"10.1137","volume":"28","author":[{"given":"Robert C.","family":"Kirby","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anders","family":"Logg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L. Ridgway","family":"Scott","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andy R.","family":"Terrel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"B. Bagheri and R. Scott,\n                      Analysa\n                      , http:\/\/people.cs.uchicago.edu\/\u223cridg\/al\/aa.html."},{"key":"R2","unstructured":"W. Bangerth, R. Hartmann, and G. Kanschat,\n                      deal.II\n                      Differential Equations Analysis Library, http:\/\/www.dealii.org."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R4","unstructured":"P. G. Ciarlet,\n                      Numerical Analysis of the Finite Element Method\n                      , Les Presses de l\u2019Universite de Montreal, Montreal, QC, Canada, 1976."},{"key":"R5","unstructured":"P. Dular and C. Geuzaine,\n                      GetDP: A General Environment for the Treatment of Discrete Problems\n                      , http:\/\/www.geuz.org\/getdp\/."},{"key":"R6","unstructured":"J. Hoffman, J. Jansson, and A. Logg,\n                      DOLFIN\n                      , http:\/\/www.fenics.org\/dolfin\/."},{"key":"R7","unstructured":"J. Hoffman and A. Logg,\n                      DOLFIN: Dynamic Object Oriented Library for FINite Element Computation\n                      , Tech. report 2002\u201306, Chalmers Finite Element Center Preprint Series, Chalmers Finite Element Center, G\u00f6teborg, Sweden, 2002."},{"key":"R8","volume-title":"Spectral\/ element methods for CFD","author":"Karniadakis George","year":"1999"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1145\/1039813.1039820"},{"key":"R10","unstructured":"R. C. Kirby,\n                      Optimizing FIAT with the level 3 BLAS\n                      , ACM Trans. Math. Software, to appear."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/040607824"},{"key":"R12","unstructured":"R. C. Kirby, M. Knepley, and L. R. Scott,\n                      Evaluation of the Action of Finite Element Operators\n                      , Tech. report 2004\u201307, University of Chicago, Chicago, IL, 2004."},{"key":"R13","unstructured":"R. C. Kirby and A. Logg,\n                      A compiler for variational forms\n                      , ACM Trans. Math. Software, to appear."},{"key":"R14","unstructured":"A. Levitin,\n                      Introduction to the Design and Analysis of Algorithms\n                      , Addison\u2013Wesley, Boston, 2003."},{"key":"R15","unstructured":"A. Logg,\n                      The FEniCS Form Compiler FFC\n                      , http:\/\/www.fenics.org\/ffc\/."},{"key":"R16","unstructured":"K. Long,\n                      Sundance\n                      , http:\/\/csmr.ca.sandia.gov\/\u223ckrlong\/sundance.html."},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Kevin Long, Sundance rapid prototyping tool for parallel PDE optimization, Lect. Notes Comput. Sci. 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Lecture Notes in Math., Vol. 60658:3547","DOI":"10.1007\/BFb0064470"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431752-8"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050635547","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:44:30Z","timestamp":1787334270000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050635547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/050635547"],"URL":"https:\/\/doi.org\/10.1137\/050635547","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}