A Prolegomenon to Differential Equations and Variational Methods on Graphs

by Yves van Gennip, Jeremy Budd

Overview

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.

Product Details

ISBN-13:	9781009346634
Publisher:	Cambridge University Press
Publication date:	02/27/2025
Series:	Elements in Non-local Data Interactions: Foundations and Applications
Pages:	100
Product dimensions:	5.98(w) x 9.02(h) x 0.20(d)

Table of Contents

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.