Title: Nonlocal models in computational science and engineering: challenges and applications.

Abstract: Nonlocal models such as peridynamics and fractional equations can capture effects that classical partial differential equations fail to capture. These effect include multiscale behavior, discontinuities in the solutions such as cracks, and anomalous behavior such as super- and sub-diffusion. For this reason, they provide an improved predictive capability for a large class of engineering and scientific applications including fracture mechanics, subsurface flow, turbulence, plasma dynamics, and image processing, to mention a few. 

However, the improved accuracy of nonlocal formulations comes at the price of modeling and computational challenges that may hinder the usability of these models. Challenges include the prescription of nonlocal boundary conditions, the treatment of nonlocal interfaces, the identification of model parameters and the incredibly high computational cost. In this talk I will discuss these challenges and describe in detail how we are addressing some of them at Sandia National Labs.

Specifically, I will describe a new flexible and physically consistent strategy for the prescription of nonlocal boundary conditions, where the challenge comes from the fact that available data are not enough for the well posedness of the nonlocal model.

If time allows, I will also briefly describe a recently developed machine learning approach to the estimation of model parameters in nonlocal diffusion models based on physics informed neural networks.