A Reduced Basis Method For Fractional Diffusion Operators I

Abstract

We propose and analyze new numerical methods to evaluate fractional norms and apply fractional powers of elliptic operators. By means of a reduced basis method, we project to a small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations of $(\text{I}-t_i^2\Delta {)}^{-1}f$, which can be computed in parallel. We prove exponential convergence rates for the optimal choice of sampling points $t_i$, provided by the so-called Zolotarëv points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm.