# Direct and Inverse Problems for PDEs with Random Coefficients - Abstract

**Gunzburger, Max**

*Reduced-order modeling and a multiscale implementation for nonlocal diffusion and mechanics problems with random inputs*

Nonlocal problems, including those involving fractional derivative operators, are characterized by interactions between points that are a finite distance apart. As a result, the corresponding finite element stiffness matrices are denser than what is encountered for PDEs. Thus, the need for reduced-order modeling for such problems is even more acute than it is for the PDE setting. We apply a reduced-order model for the computation of quantities of interest that depend on the solution of nonlocal models and for the construction of interpolatory surrogates for those solutions. We also briefly discuss means for reducing the density of the stiffness matrices through an adaptive grid refinement process.