# Fast algorithm for computing nonlocal operators with finite interaction distance

## TL;DR

This paper introduces a novel fast algorithm for nonlocal operators with finite interaction distance, significantly reducing computational costs and enabling more efficient simulations of nonlocal phenomena.

## Contribution

A new class of fast algorithms is developed for nonlocal operators, overcoming previous limitations and matching the efficiency of local numerical methods in some cases.

## Key findings

- Reduces computational complexity of nonlocal operators
- Applicable to models like peridynamics and fractional diffusion
- Enables efficient large-scale simulations

## Abstract

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.