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T. Danczul, J. Schöberl:
"A reduced basis method for fractional diffusion operators II";
Journal of Numerical Mathematics, 29 (2021), 4; 269 - 287.

We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛-sf to fractional PDEs involving elliptic operators. Reinterpreting 𝓛-s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. 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. 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 the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously.

A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.

http://dx.doi.org/10.1515/jnma-2020-0042