# Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models

## TL;DR

This paper develops a reduced basis method for parametrized nonlocal and fractional Laplace models, addressing challenges of non-affine, singular kernels, and variable regularity, with theoretical and numerical validation.

## Contribution

It introduces an affine approximation technique for complex parametrized kernels and provides a comprehensive error analysis for model order reduction.

## Key findings

- Effective affine kernel approximations achieved
- Reliable a posteriori error estimators developed
- Numerical experiments confirm theoretical results

## Abstract

We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. In order to provide an efficient and reliable approximation of the solution for different values of the parameters, we develop the reduced basis method as a parametric model order reduction approach. Major difficulties arise since the kernels are not affine in the parameters, singular, and discontinuous. Moreover, the spatial regularity of the solutions depends on the varying fractional power $s$. To address this, we derive regularity and differentiability results with respect to $\delta$ and $s$, which are of independent interest for other applications such as optimization and parameter identification. We then use these results to construct affine approximations of the kernels by local polynomials. Finally, we certify the method by providing reliable a posteriori error estimators, which account for all approximation errors, and support the theoretical findings by numerical experiments.