An adaptive radial basis function approach for efficiently solving multidimensional spatiotemporal integrodifferential equations

TL;DR

This paper introduces an adaptive radial basis function method that efficiently solves complex multidimensional spatiotemporal integrodifferential equations by automatically adjusting RBF parameters to handle high-dimensional problems.

Contribution

The paper presents a novel adaptive RBF approach that automatically adjusts shape parameters, improving efficiency and ease of implementation for multidimensional spatiotemporal equations.

Findings

Effectively mitigates the curse of dimensionality.

Demonstrates high accuracy in numerical examples.

Provides a mesh-free, easy-to-implement solution method.

Abstract

In this work, we propose an adaptive radial basis function (RBF) approach for the efficient solution of multidimensional spatiotemporal integrodifferential equations. Our approach can automatically adjust the shape of RBFs and provide an easy-to-implement and mesh-free approach for solving spatiotemporal equations. Specifically, we analyze how the proposed method mitigates the curse of dimensionality by adaptively adjusting the scales and centers of the radial basis functions when the solution is spatially anisotropic. Through a range of numerical examples, we demonstrate the effectiveness of our approach for solving multidimensional spatiotemporal integrodifferential equations.