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

The book presents the state of the art of nonlocal modeling and discretization and provides a practical introduction to nonlocal modeling for readers who are not familiar with such 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 thus allow for the capture of multiscale effects, the result of which is their successful use in many scientific and engineering applications. The book also provides a thorough analysis and numerical treatment of nonstandard nonlocal models, focusing on both well-known and nonstandard interaction neighborhoods. In addition, the book de-livers an extensive practical treatment of the implementation of discretization strategies via finite element methods. Numerous figures are provided as concrete examples to illustrate both the analytic and computational results.

Marta D'Elia is a principal scientist and lead of scientific machine learning at Pasteur Labs and an adjunct professor at Stanford in the Institute for Computational and 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 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 O. Lawton Distinguished Professor/Marie Krafft Professor at Florida State University and Senior Research Fellow at the University of Texas at Austin. He is a founding editor of ASA/SIAM Journal on Uncertainty Quantification and served as editor-in-chief of SIAM Journal on Numerical Analysis. His research, which includes finite element methods, optimization and control, uncertainty quantification, multifidelity methods, nonlocal modeling, reduced-order methods, applied analyses, and computational geometry, has significantly impacted several application 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. He was 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.