# Approximation of Definite Integrals Over the Volume of the Ball

## TL;DR

This paper introduces a novel RBF-FD inspired method for efficiently computing definite integrals over 3D ball volumes using arbitrarily scattered nodes, avoiding uniformity constraints and achieving high accuracy with optimal computational complexity.

## Contribution

The paper presents a new RBF-FD based algorithm that computes quadrature weights for scattered nodes in three dimensions without uniformity restrictions, with improved efficiency.

## Key findings

- Computes quadrature weights in O(N log N) operations.
- Achieves high-order accuracy for scattered nodes.
- Removes the need for uniform node distribution.

## Abstract

A Radial Basis Function Generated Finite-Differences (RBF-FD) inspired technique for evaluating definite integrals over the volume of the ball in three dimensions is described. Such methods are necessary in many areas of Applied Mathematics, Mathematical Physics and many other application areas. Previous approaches needed restrictive uniformity in the node set, which the algorithm presented here does not require. By using RBF-FD approach, the proposed algorithm computes quadrature weights for $N$ arbitrarily scattered nodes in only $O(N\mbox{ log}N)$ operations with high orders of accuracy.