A Prolegomenon to Differential Equations and Variational Methods on Graphs

A bird's eye view of theoretical developments and concrete applications that will enable newcomers to get a flavour of key results.

Yves van Gennip (Author), Jeremy Budd (Author)

9781009346634, Cambridge University Press

Paperback / softback, published 27 February 2025

100 pages
22.8 x 15.1 x 0.1 cm, 0.16 kg

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.

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.

Subject Areas: Numerical analysis [PBKS]