Yves van Gennip

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.

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.

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.