Pydec: Software and Algorithms for Discretization of Exterior Calculus
PyDEC: Software and Algorithms for Discretization of Exterior Calculus NATHAN BELL, Google Inc. ANIL N. HIRANI, University of Illinois at Urbana-Champaign This article describes the algorithms, features, and implementation of PyDEC, a Python library for computa- tions related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest-order finite element exterior calculus, and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the 3 facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples. Categories and Subject Descriptors: G.4 [Mathematical Software]: ; G.1.8 [Numerical Analysis]: Par- tial Differential Equations—Finite element methods, finite volume methods; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems General Terms: Algorithms, Theory, Experimentation, Design Additional Key Words and Phrases: Discrete exterior calculus, finite element exterior calculus, Whitney form, simplicial complex, cubical complex, Vietoris-Rips complex, computational topology, boundary operator, coboundary operator, chain, cochain ACM Reference Format: Bell, N. and Hirani, A. N. 2012. PyDEC: Software and algorithms for discretization of exterior calculus. ACM Trans. Math. Softw. 39, 1, Article 3 (November 2012), 41 pages. DOI = 10.1145/2382585.2382588 http://doi.acm.org/10.1145/2382585.2382588 1.
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