TITLE: Nonlocal Diffusion Problems with Inequality Constraints

SPEAKER: Olena Burkovska, Computer Science and Mathematics Division, Oak Ridge National Laboratory, TN

ABSTRACT:  In this talk we discuss regularity and approximation of nonlocal diffusion models involving variational inequalities. These problems arise in different fields such as, e.g., contact mechanics, finance, and interface models. We consider nonlocal operators with a finite range of interactions, and both singular (fractional Laplacian) as well as integrable kernels. In the first part of the talk we present the spatial and parametric regularity of the solution and associated bilinear form. Here, we consider the fractional exponent and nonlocal interaction radius to be parameters in the model, and we derive corresponding differentiability results. Exploiting these results, we demonstrate how to construct the efficient matrix assembly and approximation approach based on the reduced basis method. These developments are particularly beneficial in the multi-query context, naturally arising in, e.g., optimization and parameter identification. Additionally, we certify the method by providing reliable a posteriori error estimators. Finally, we give an overview of ongoing work on more complex problems involving nonlocal variational inequalities, in particular the Cahn-Hilliard model. While this is a diffuse interface model in the classical local setting, we demonstrate how it turns into a sharp-interface model in the nonlocal case for a careful choice of the nonlocal Ginzburg-Landau energy. Here, in contrast to the first part of the talk, we exploit the reduced regularity of the solution as well as the inclusion of the inequality constraints.