Modeling and Inverse Problems of Nonlocal Phenomena

Dear Colleagues,

Recent studies, both experimental and theoretical, have shown that certain phenomena in natural sciences, such as the foraging of ants or high-energy collisions of heavy ions, are more accurately modeled by stochastic processes with jumps rather than pathwise continuous processes. Generally, the infinitesimal generators of jump processes are nonlocal operators and non-classical differential operators, as in the case of Brownian motion. The key feature of nonlocal operators is that the function needs to be known in a typically large neighborhood in order to compute their effect on a given input function at a certain point. Studying certain stochastic games related to these jump processes leads to exterior value problems for partial differential equations (PDEs) involving nonlocal operators and other functions that represent the properties of the dynamical system. For inverse problems for nonlocal PDEs, the goal is to recover the unknown functions from indirect measurements. These unknowns can include coefficients or exponents in the nonlocal operator, lower-order perturbations, or a Riemannian metric; measurements can be encoded in a Dirichlet-to-Neumann map, source-to-solution map, or initial-to-final state map. Therefore, progress in the area of nonlocal inverse problems relies heavily on advances in the modeling of nonlocal phenomena. This Special Issue aims to bring together and highlight recent developments in the modeling of nonlocal phenomena and their associated inverse problems.

We invite submissions related (but not limited to) the following topics:

• Nonlocal partial differential equations;
• Nonlinear fractional dynamical systems;
• Anomalous diffusion models;
• Recovery of coefficients and nonlinearities in nonlocal models;
• Unique continuation property of nonlocal operators;
• Runge’s approximation;
• Stability estimates for nonlocal inverse problems;
• Applications of nonlocal models to the natural sciences.

Dr. Philipp Zimmermann
Guest Editor

Manuscript Submission Information

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• nonlocal partial differential equations
• nonlinear fractional dynamical systems
• anomalous diffusion models
• nonlocal operators
• Runge’s approximation
• nonlocal inverse problems