Dynamic Isoperimetry on Graphs and Weighted Riemannian manifolds

dc.contributor.advisor Froyland, Gary en_US Kwok, Eric en_US 2022-03-22T16:55:39Z 2022-03-22T16:55:39Z 2018 en_US
dc.description.abstract Transport and mixing in dynamical systems are important properties for many physical, chemical, biological, and engineering processes. The detection of transport barriers for dynamics with general time dependence is a difficult, but important problem, because such barriers control how rapidly different parts of phase space (which might correspond to different chemical or biological agents) interact. The key factor is the growth of interfaces that partition phase space into separate regions. In a recent paper, Froyland introduced the notion of dynamic isoperimetry: the study of sets with persistently small boundary size (the interface) relative to enclosed volume, when evolved by the dynamics. Sets with this minimal boundary size to volume ratio were identified as level sets of dominant eigenfunctions of a dynamic Laplace operator. In this dissertation, we develop a data-driven approach for transport barrier detection, by extending and generalising dynamic isoperimetry to graphs and weighted Riemannian manifolds. First we model trajectory data as dynamics of graphs. We use minimium disconnecting cuts to search for coherent structure in dynamic graphs, where the graph dynamic arises from a general sequence of vertex permutations. We develop a dynamic spectral partitioning method via a new dynamic Laplacian matrix. We prove a dynamic Cheeger inequality for graphs, and demonstrate the effectiveness of this dynamic spectral partitioning method on both structured and unstructured graphs. We then generalise the dynamic isoperimetric problem on manifolds to situations where the dynamics (i) is not necessarily volume-preserving, (ii) acts on initial agent concentrations different from uniform concentrations, and (iii) occurs on a possibly curved phase space. Our main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, the dynamic Cheeger's inequality, and the Federer-Fleming theorem. We illustrate the computational approach with some simple numerical examples. Finally, we form a connection between the weighted graph version of our dynamic Laplacian matrix and the manifold dynamic Laplace operator. We then form a dynamic Laplacian-based manifold learning algorithm, which is designed to approximate solutions of our generalised dynamic isoperimetric problem from trajectory data. We highlight the robustness of our dynamic manifold learning method through numerical experiments. en_US
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri en_US
dc.subject.other Lagrangian coherent structure en_US
dc.subject.other Dynamic en_US
dc.subject.other Isoperimetry en_US
dc.subject.other Weighted manifold en_US
dc.subject.other Graph en_US
dc.subject.other Manifold learning en_US
dc.title Dynamic Isoperimetry on Graphs and Weighted Riemannian manifolds en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Kwok, Eric
dspace.entity.type Publication en_US
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Kwok, Eric, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Froyland, Gary, Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
public version.pdf
5.57 MB
Resource type