Differential Equations and Variational Methods on Graphs

A detailed introduction to differential equations on graphs, a field rich with mathematical challenges and abundant applications.

A detailed overview of differential equations on graphs, suitable for researchers and graduate students in mathematical image analysis, mathematical machine learning, and at the interface of calculus of variations and optimisation. Clearly explaining the basics, and covering diverse applications, this is the perfect introduction to a rich field.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.