A Prolegomenon to Differential Equations and Variational Methods on Graphs

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Yves van Gennip (Delft University of Technology) Jeremy Budd (California Institute of Technology)

A Prolegomenon to Differential Equations and Variational Methods on Graphs

Yves van Gennip (Delft University of Technology), Jeremy Budd (California Institute of Technology)

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English

Cambridge University Press
27 February 2025

Numerical analysis

Series: Elements in Non-local Data Interactions: Foundations and Applications

The use of differential equations on graphs as a framework for the mathematical analysis of images emerged about fifteen years ago and since then it has burgeoned, and with applications also to machine learning. The authors have written a bird's eye view of theoretical developments that will enable newcomers to quickly get a flavour of key results and ideas. Additionally, they provide an substantial bibliography which will point readers to where fuller details and other directions can be explored. This title is also available as open access on Cambridge Core.

By: Yves van Gennip (Delft University of Technology), Jeremy Budd (California Institute of Technology)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
ISBN: 9781009494656
ISBN 10: 1009494651
Series: Elements in Non-local Data Interactions: Foundations and Applications
Pages: 100
Publication Date: 27 February 2025
Audience: General/trade , ELT Advanced
Format: Hardback
Publisher's Status: Active

1. Introduction; 2. History and literature overview; 3. Calculus on undirected edge-weighted graphs; 4. Directed graphs; 5. The graph Ginzburg–Landau functional; 6. Spectrum of the graph Laplacians; 7. Gradient flow: Allen–Cahn; 8. Merriman–Bence–Osher scheme; 9. Graph curvature and mean curvature flow; 10. Freezing of Allen–Cahn, MBO, and mean curvature flow; 11. Multiclass extensions; 12. Laplacian learning and Poisson learning; 13. Conclusions; Bibliography.