Published: Jan. 18, 2019

Title: "Approximation of parametrized kernels arising in nonlocal and fractional Laplace models"

Abstract: We consider parametrized linear and obstacle problems driven by a spatially nonlocal integral operator. These problems have a broad impact on current developments in different fields such as, e.g., peridynamics, contact mechanics, and finance. We focus on integral kernels with nonlocal interactions limited to a ball of radius greater than 0 or (truncated) fractional Laplace kernels, which are also parametrized by the fractional power s ∈ (0,1). Compared to  the fractional problems with infinite horizon of interaction,  these type of problems are of independent interest, since they form a connection between purely nonlocal and classical local PDE problems. Our goal is to provide an efficient and reliable approximation of the solution for different values of the kernel parameters. To reduce the high computational cost associated with multi-query solution evaluations, we employ the reduced basis method (RBM) as a parametric model order reduction approach. A major difficulty in the construction of the method arises in the non-affinity of the integral kernel w.r.t. the parameters, which can not be directly treated by empirical interpolation due to the singularity and a lack of continuity of the kernel. This substantially affects the efficiency of the RBM. As a remedy, we propose suitable approximations of the kernel, based on the parametric regularity of the bilinear form and the improved spatial regularity of the solution. The results we provide are of independent interest for other approximation techniques and applications such as, e.g., optimization or parameter identification. Finally, we certify the RBM by providing reliable a posteriori error estimators and support the theoretical findings by numerical experiments.