# Model reduction for fractional elliptic problems using Kato's formula

## TL;DR

This paper introduces a stable, efficient model reduction algorithm for solving fractional elliptic PDEs using Kato's formula, significantly reducing computational costs for multiple queries.

## Contribution

It develops a novel, stable discretization of the integral representation of fractional Laplacian solutions combined with reduced basis methods for efficient multi-query computations.

## Key findings

- Stable discretization independent of discretization parameters
- Gaussian quadrature improves efficiency over previous methods
- Order of magnitude reduction in computational cost for multiple queries

## Abstract

We propose a novel numerical algorithm utilizing model reduction for computing solutions to stationary partial differential equations involving the spectral fractional Laplacian. Our approach utilizes a known characterization of the solution in terms of an integral of solutions to classical elliptic problems. We reformulate this integral into an expression whose continuous and discrete formulations are stable; the discrete formulations are stable independent of all discretization parameters. We subsequently apply the reduced basis method to accomplish model order reduction for the integrand. Our choice of quadrature in discretization of the integral is a global Gaussian quadrature rule that we observe is more efficient than previously proposed quadrature rules. Finally, the model reduction approach enables one to compute solutions to multi-query fractional Laplace problems with order of magnitude less cost than a traditional solver.