Nonlocal Modeling, Analysis, and Computation

Studies of complexity, singularity, and anomaly using nonlocal continuum models are steadily gaining popularity. This monograph provides an introduction to basic analytical, computational, and modeling issues and to some of the latest developments in these areas. Nonlocal Modeling, Analysis, and Computation includes motivational examples of nonlocal models, basic building blocks of nonlocal vector calculus, elements of theory for well-posedness and nonlocal spaces, connections to and coupling with local models, convergence and compatibility of numerical approximations, and various applications, such as nonlocal dynamics of anomalous diffusion and nonlocal peridynamic models of elasticity and fracture mechanics. A particular focus is on nonlocal systems with a finite range of interaction to illustrate their connection to local partial differential equations and fractional PDEs. These models are designed to represent nonlocal interactions explicitly and to remain valid for complex systems involving possible singular solutions and they have the potential to be alternatives for as well as bridges to existing models. The author discusses ongoing studies of nonlocal models to encourage the discovery of new mathematical theory for nonlocal continuum models and offer new perspectives on traditional models, analytical techniques, and algorithms. Nonlocal Modeling, Analysis, and Computation offers applied mathematicians, computational scientists, researchers, and graduate students an illustration of the broad applicability and rich mathematics of nonlocal models and provides motivation for further investigations.

Qiang Du is the Fu Foundation Professor of Applied Mathematics at Columbia University, where he chairs the applied mathematics program in the Department of Applied Physics and Applied Mathematics and is a co-chair of the Center for Foundations of Data Science in the Data Science Institute. Professor Du was the Verne M. Willaman Professor of Mathematics and Professor of Materials Science and Engineering at Penn State University. Recognition for his work includes the SIAM Outstanding Paper Prize and ACM Gordon Bell Prize Finalist in 2016. He was an invited speaker at the International Congress of Mathematicians (ICM 2018) and is a Fellow of SIAM and of the American Association for the Advancement of Science.