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Topological optimization of the evaluation of finite element matrices

Journal article
Authors R.C. Kirby
Anders Logg
L. Ridgway Scott
A.R. Terrel
Published in SIAM Journal on Scientific Computing
Volume 28
Issue 1
Pages 224-240
ISSN 1064-8275
Publication year 2006
Published at Department of Mathematical Sciences, Mathematics
Pages 224-240
Language en
Links dx.doi.org/10.1137/050635547
Keywords Finite element , Minimum spanning tree , Optimized algorithm , Variational form
Subject categories Mathematics, Computational Mathematics

Abstract

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. © 2006 Society for Industrial and Applied Mathematics.

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