Applied Analysis Seminar - Abstracts
Talk
Thursday, 24 October 2019, 13:30-14:30 in HG03.085
Yves van Gennip (TU Delft)
Graph Ginzburg--Landau: discrete dynamics, continuum limits, and applications. An overview.
Abstract
Applications that can be described by variational models profit from all the advantages those models bring
along. Both on the functional level as on the level of the associated differential equations,
powerful techniques have been developed over the years to study these models. Up until fairly
recently, such models were typically formulated in a continuum setting, i.e. as the minimization
of a functional over an admissible class of functions whose domains are subsets of Euclidean space
or Riemannian manifolds. The field of variational methods and partial differential equations (PDEs)
on graphs aims to harness the power of variational methods and PDEs to tackle problems that
inherently have a graph (network) structure.
In this talk we will encounter the graph Ginzburg--Landau model, which is a paradigmatic
example of a variational model on graphs. Just as its continuum forebear is used to model
phase separation on a continuum domain ---it assigns to each point of the domain a value
from an (approximately) discrete set of values--- the graph Ginzburg--Landau model describes
phase separation on the nodes of a graph. This makes it extremely well suited for applications
such as data clustering, data classification, community detection in networks, and image segmentation.
Theoretically there are also interesting questions to ask, often driven by the properties
that have already been established for the continuum Ginzburg--Landau model, such as
Gamma-convergence properties of the functional and relationships between its associated
differential equations. This presentation will give an overview of some recent developments.
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