On the approximation of spatial convolutions by PDE systems

TL;DR

This paper develops a PDE-based approximation method for spatial convolutions with radial kernels in higher dimensions, using Green functions and integral transformations, validated by numerical examples.

Contribution

It introduces a PDE system approximation for multidimensional spatial convolutions with arbitrary radial kernels, expanding previous 1D approaches.

Findings

Effective approximation of radial convolutions in higher dimensions

Green functions form a complete basis for the approximation

Numerical examples demonstrate the method's accuracy

Abstract

This paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can eliminate the analytical difficulties arising from integral formulations in one-dimensional space. In this paper, we establish a PDE system approximation for spatial convolutions in higher spatial dimensions. We derive an appropriate approximation function for given arbitrary radial integral kernels as a linear sum of Green functions. In establishing the validity of this methodology, we introduce an appropriate integral transformation to show the completeness of the basis constructed by the Green functions. This framework enables the approximation of nonlocal convolution-type operators with arbitrary radial integral kernels using linear sums of PDE solutions. Finally, we present numerical examples that illustrate the effectiveness of our proposed method.