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

Abstract:ÌýWe consider parametrized linear and obstacle problemsÌýdriven by a spatially nonlocal integral operator. These problems have a broad impact on current developments in different fields such as, e.g.,Ìýperidynamics, contact mechanics, and finance. We focus on integral kernels with nonlocal interactionsÌýlimited to a ball of radius greater thanÌý0 orÌý(truncated)Ìýfractional Laplace kernels, which are alsoÌýparametrizedÌýby the fractional power s ∈ (0,1). Compared toÌý the fractional problems with infinite horizon of interaction,Ìý theseÌýtype of problems are of independent interest, since they form a connection betweenÌýpurely nonlocal and classical local PDEÌýproblems. Our goal is to provide an efficient and reliableÌýapproximation 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 andÌýa lack of continuity of the kernel. ThisÌýsubstantially 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 regularityÌýof the solution. The results we provide are of independent interest for other approximation techniquesÌýandÌýapplications such as, e.g., optimization or parameter identification. Finally, we certify the RBMÌýby providing reliable a posteriori error estimators and support the theoretical findings by numericalÌýexperiments.