A Reduced Basis Method For Fractional Diffusion Operators II

We present a novel numerical scheme to approximate the solution map $s\mapsto u(s) := $\mathcal{L}$^{-s}f$ to partial differential equations involving fractional elliptic operators. Reinterpreting ${\mathcal{L}}^{-s}$ as interpolation operator allows us to derive an integral representation of $u(s)$ which includes solutions to parametrized reaction-diffusion problems. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. Avoiding further discretization, the integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation $L$ of the operator whose inverse is projected to a low-dimensional space, where explicit diagonalization is feasible. The universal character of the underlying $s$-independent reduced space allows the approximation of $(u(s){)}_{s\in (0,1)}$ in its entirety. We prove exponential convergence rates and confirm the analysis with a variety of numerical examples. Further improvements are proposed in the second part of this investigation to avoid inversion of $L$. Instead, we directly project the matrix to the reduced space, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.