Differential Equations and Variational Methods on Graphs: With Applications to Machine Learning and Image Analysis (Cambridge Monographs on Applied and Computational Mathematics, Series Number 44)

The burgeoning field of differential equations on graphs has experienced significant growth in the past decade, propelled by the use of variational methods in imaging and by its applications in machine learning. This text provides a detailed overview of the subject, serving as a reference for researchers and as an introduction for graduate students wishing to get up to speed. The authors look through the lens of variational calculus and differential equations, with a particular focus on graph-Laplacian-based models and the graph Ginzburg-Landau functional. They explore the diverse applications, numerical challenges, and theoretical foundations of these models. A meticulously curated bibliography comprising approximately 800 references helps to contextualise this work within the broader academic landscape. While primarily a review, this text also incorporates some original research, extending or refining existing results and methods.

Yves van Gennip is Associate Professor in the Delft Institute of Applied Mathematics (DIAM) at Delft University of Technology.

Jeremy Budd is Assistant Professor of Mathematics and its Applications at the University of Birmingham.