A Reduced Basis Method For Fractional Diffusion Operators I

TL;DR

This paper introduces a reduced basis method for efficiently computing fractional powers of elliptic operators, leveraging parallelizable inverse evaluations and Zolotar"ev points for exponential convergence.

Contribution

It presents a novel reduced basis approach with proven exponential convergence for fractional diffusion operators, enabling efficient parallel computations.

Findings

Exponential convergence rates achieved with optimal sampling points.

Numerical experiments confirm the efficiency and accuracy of the method.

Parallelizable evaluations improve computational performance.

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 $(\textrm{I}-t_i^2\Delta)^{-1}f$, 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 Zolotar\"ev points. Numerical experiments confirm the analysis and demonstrate the efficiency of our algorithm.