Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) c...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 14; no. 1; pp. 151 - 184
Main Authors: Harker, Shaun, Mischaikow, Konstantin, Mrozek, Marian, Nanda, Vidit
Format: Journal Article
Language:English
Published: Boston Springer US 01.02.2014
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Computational mathematics
Computer Science
Economics
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Theory
55-04
Computational homology
Discrete Morse theory
57-04
55N35
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary:We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such set-valued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.
Bibliography:SourceType-Scholarly Journals-1
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content type line 14
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-013-9145-0