Nonlocality: Analysis, Numerics and Applications

This workshop aims to bring together leading experts and young researchers interested in nonlocal problems. The thematic range covers aspects of nonlocality in four closely interacting areas of mathematics: Analysis of Partial Differential Equations, Calculus of Variations, Numerics and Scientific Computing, Modeling and Applications.

Traditionally, many problems in science and engineering are modeled by local partial differential equations (PDEs) or variational principles. Locality in this context means that the behavior of an object depends solely on its immediate neighborhood. It is, however, evident that long-range and global effects occur naturally in many situations and cannot be neglected. The latter gives rise to the nonlocal models. Among the many examples of applications are peridynamics, dislocation theory, geophysics, pattern formation, micromagnetics, diffusion processes, population dispersal, phase transitions, image processing, deep neural networks, etc.

Compared to the classical, meaning local, theories in analysis, many new conceptual challenges arise for nonlocal models - even in what may appear to be rather basic questions at first. In fact, boundary values, for instance, are inherently local objects – a number of possible nonlocal replacements exist, and they are not usually equivalent. The question of the existence of minimizers of local energy functionals can be answered using well-established tools. In the nonlocal setting, however, many of these techniques, which build on localization principles, are nonexistent. The difficulty due to nonlocality is not limited to modeling or analysis, but it extends to the development of efficient numerical algorithms. For instance, evaluating a nonlocal operator at a given point requires taking into account the much larger physical domain and not just the closest neighboring point. One is confronted with approximate singular integrals and has to deal with dense linear algebra. Thus, naive approaches very quickly become untenable.

Motivated by such challenges, there has been substantial progress in the mathematical study of nonlocal problems over the last decade. Let us mention here just three exemplary achievements from the fields mentioned above: viscosity solution techniques have been adapted to equations involving fractional differential operators; criteria for lower-semicontinuity of certain classes of nonlocal functionals have been identified using refined Young measure techniques; multiscale algorithms and using the characterization of fractional Laplacians via extension problems are often feasible numerical approaches.

This workshop will provide a fertile ground for exchanging ideas about these novel analytical and numerical concepts. It aims to identify new research directions for the next decade. Above all, it will provide a unique opportunity to young scientists and help create the next generation of scientists.