Nonlocal Integral Equation Continuum Models: Nonstandard Interaction Domains and Finite Element Discretizations

The book presents the state of the art of nonlocal modeling and discretization and novel analyses of a class of nonstandard nonlocal models. These models have recently become a viable alternative to classical partial differential equations when the latter are unable to capture effects such as discontinuities and multiscale behavior in a system of interest. Because of their integral nature, nonlocal operators allow for the relaxation of regularity requirements on the solution and for capturing multiscale effects and thus have been successfully used in scientific and engineering applications such as diffusion processes, fracture mechanics, heterogeneous material response, subsurface transport, turbulence, and image processing, to name a few. Although the use of nonstandard models is novel, the book provides extensive background and a thorough analysis and description of their discretization methods, enabling it to serve as a gentle and practical introduction to nonlocal modeling for readers who are not familiar with nonlocality.

About the Author :
Marta D'Elia is a Principal Scientist at Pasteur Labs and an Adjunct Professor at Stanford ICME (Institute for Computational & Mathematical Engineering). She previously worked at Meta as a research scientist and at Sandia National Laboratories as a principal member of the technical staff. She is an Associate Editor of several journals, including SIAM Journal on Scientific Computing, a member of several committees, including the SIAM Industry Committee, and is vice chair of the Northern and Central California Section of SIAM. Her work focuses on the design and analysis of machine learning models and data-driven algorithms for the simulation of complex, multiscale, and multiphysics problems. Max Gunzburger is emeritus Robert Lawton and Marie Krafft Professors at Florida State University and Senior Research Fellow at the University of Texas at Austin. He has served as a founding editor the ASA/SIAM JUQ journal and as editor in chief of SINUM. His research includes, among other topics, finite element methods, optimization and control, uncertainty quantification, multifidelity methods, nonlocal modeling, reduced-order methods, applied analyses, and computational geometry and has also impacted several applications areas such as fluid and solid mechanics, superconductivity, and electromagnetics. Christian Vollmann works as a software engineer at Tesla Automation with a focus on computer vision and machine learning, and he held a position as Akademischer Rat at Trier University from 2020 to 2023. His research includes the development of finite element methods for nonlocal models, the analysis of nonlocal boundary value problems, and shape optimization.