Year: 2015
Type: article
Abstract: Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function f : X → R f \colon X \to \mathbb {R} , where X X is a CW-complex. In the special case X = [ 0 , 1 ] N X = [0,1]^N , N ∈ N N \in \mathbb {N} , we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.
Source: Mathematics of Computation
DOI: 10.1090/mcom/3137
Authors: Jonathan Jaquette, Miroslav Kramár
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
Awards: G1389074705, G1523888516, CDI-Type II: Collaborative Research: Computational Homology, Jamming, and Force Chains in Dense Granular Flows, G3442010469, Collaborative proposal: Computing Dynamics of Multiparameter Systems, G3999492190, G4868937273, G5222733069, CDI-TYPE II--COLLABORATIVE RESEARCH: Using Algebraic Topology to Connect Models with Measurements in Complex Nonequilibrium Systems
OpenAlex ID: W2252892898