Qiang Du
SIAM 2019, 163 PAGES
PRICE (PAPERBACK) £53.50 ISBN 978-1-61197-561-1
Our world is getting more and more nonlocal: what happens at a given time and place is being increasingly influenced by events at extended distances and at significantly earlier times. Consequently, nonlocal models are growing in importance. As the power of computing technology grows, they are also becoming more tractable. These factors mean that the release of this book is especially timely.
The text describes progress on the development of a rigorous and systematic mathematical foundation of nonlocal models. These theoretical considerations are balanced with descriptions of how to conduct effective and robust numerical simulation. In addition, the text helpfully discusses both connections to local models and the distinctions present in nonlocal models. This usefully places nonlocal models as both complements and alternatives to local models.
The first two chapters provide introductory material, firstly to the book and its organisation, and secondly to the topic of nonlocal modelling. The topic-based introduction clearly describes how nonlocality appears from physical examples, which naturally motivates modelling of such systems. This introduction also provides a history of the field, placing current work as an extension of earlier efforts, including local models.
The next three chapters consider specific aspects of nonlocal models that may be most relevant to specific audiences. For example, the third chapter presents the mathematical framework of nonlocal models. It includes an overview of nonlocal vector calculus, including a helpful table that compares nonlocal elements with their better-known counterparts in local calculus: things like differential operators and Green’s identity are covered, for example. This chapter also includes detailed introductions to some nonlocal spaces. Consequently, this chapter is targeted towards readers who are interested in the rigorous analysis of the mathematical theory of nonlocal models.
Being devoted to the numerical discretisation of nonlocal models, the fourth chapter is targeted towards readers interested in numerical solution and computer simulation. It begins, like all of the chapters, with a helpful quote, which situates the upcoming text for the reader. The opening quote for this chapter notes that, ‘the possibility of numerical approximations converging to a wrong solution is alarming; without prior knowledge, such convergence might be mistakenly used to verify or approve simulation results’. This demonstrates the author’s practical experience, which is helpfully described throughout the book to the benefit of the reader.
The fifth chapter covers the coupling of local and nonlocal models, which is likely to be helpful for researchers wishing to apply nonlocal approaches to multiscale and multiphysics problems. A large portion of the book is concerned with systems that are nonlocal in space. However, balance is provided by the sixth chapter, which discusses nonlocal-in-time models with finite memory effects. This provides an opportunity to appreciate the differences and similarities between the different forms of nonlocality. One obvious difference between these two cases is that time is, generally speaking, irreversible.
The seventh, and final, chapter neatly concludes the book. It includes brief discussion of additional topics. It also highlights open questions and continuing challenges, including, from a computational perspective, discretization and efficient solvers. This provides a gateway for researchers to enter the world of nonlocal modelling, analysis and computation.
Within the text, long and technical derivations are kept to a minimum. For example, proofs are sketched rather than being described in great detail. This approach works well, making the text accessible and relatively fast-paced. Should more detail be needed then this will almost certainly be found in one of the numerous references, which are provided.
To maintain the brevity provided by, for example, sketch proofs, a certain amount of background knowledge is assumed. Examples include ‘the standard second order central difference operator associated with the grid’ and ‘basic exposure to applied functional analysis and Sobolev spaces’. Assuming this knowledge means that the text is tightly focused.
Overall, this book provides a great introduction to a topic of growing importance. The targeting of particular chapters to specific audiences means that it should be appreciated by a wide range of mathematicians.
Rob Ashmore CMath CSci FIMA
Defence Science and Technology Laboratory
The views and opinions expressed herein are those of the author and do not necessarily reflect those of the Defence Science and Technology Laboratory.
Book review published directly onto IMA website August 2021