# Reduced Basis Approximations of the Solutions to Spectral Fractional Diffusion Problems

## TL;DR

This paper introduces a reduced basis method to efficiently approximate solutions to spectral fractional diffusion problems, significantly reducing computational costs while maintaining accuracy across different fractional powers.

## Contribution

The work develops a fractional power-independent reduced basis strategy for spectral fractional diffusion, enabling fast online evaluations with proven exponential convergence.

## Key findings

- Reduced basis method accelerates reaction-diffusion problem solutions.
- The approach is effective uniformly over fractional powers in [s_min, s_max].
- Numerical experiments confirm exponential convergence.

## Abstract

We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction-diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction-diffusion problems. The reduced basis does not depend on the fractional power $s$ for $0<s_{\min}\leq s \leq s_{\max}<1$. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.