Nonlocal Modeling, Analysis, and Computation

The front matter includes the title page, series page, copyright page, dedication, TOC, foreward, and preface.

Our perceived world is effectively getting more and more nonlocal: what happens in a particular position and at a specific time is seen to be increasingly affected by events in other positions over extended distances and at different times.

Nonlocal connections are ubiquitous in nature. To offer a flavor of how nonlocality enters into mathematical modeling, we begin our discussion with a brief account of various modeling choices. We then give a few illustrative examples of nonlocal models and operators and make comparisons with other traditional models. We also mention some related mathematical and computational issues to motivate more detailed studies in later chapters.

Much of the mathematical theory for continuum models represented by PDEs has its foundation in the classical (local) vector calculus originating with Newton and Leibniz and the subsequent development of exterior calculus, differential geometry, calculus of variations, and analysis of differential operators defined in function spaces. We are interested in the systematic development of the nonlocal analogue of the local theory based on the traditional vector calculus for differential operators. The new nonlocal framework is designed for nonlocal models using integral operators. In particular, it focuses on dealing with nonlocal interactions with scale- (horizon-) dependent kernels. Our study touches on the connections between the nonlocal theory and the local counterparts (seen as the local limit of the former) and on the differences between them. The latter are given particular emphasis so that one does not treat the nonlocal theory as a carbon copy of the classical theory but rather a study with distinct features. Moreover, by looking at the problems from a different angle (and with the nonlocal interaction offering an extra dimension), we can gain new insight into the study of traditional local models.

As most of the nonlocal models of practical interest cannot be solved exactly, their numerical approximation has thus attracted much attention.

Nonlocal models can be effective alternatives to the traditional local continuum model by allowing possibly singular solutions in function spaces larger than their local counterparts. This feature makes nonlocal models particularly appealing in subjects such as fracture mechanics. However, treating nonlocality in numerical simulations could incur higher computational costs in general. Thus, localization, wherever the local limits can be utilized, and effective coupling between nonlocal models and localized ones can both be very useful in practice. Nevertheless, nonlocal models generically employ nonlocal constraints near the boundary or interface, in contrast to local models, where local boundary or interface conditions can be imposed on a lower dimensional surface. This suggests several possible approaches for local and nonlocal modeling.

Much of the discussion in earlier chapters of this monograph is concerned with nonlocal spatial interactions. From the viewpoint of model reduction, spatial nonlocality is often accompanied by temporal correlations and memory effects. The latter involve nonlocality in time, the focus of our study in this chapter.

Nonlocal models, arguably more general than their local or discrete analogues, are designed to account for nonlocal interactions explicitly and to remain valid for complex systems involving possibly singular solutions. They can be alternatives and bridges to existing local continuum and discrete models. Their increasing popularity in applications makes the development of a systematic/axiomatic mathematical framework for nonlocal models necessary and timely.

The back matter includes bibliography, index, and back cover.