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On the approximation of spatial convolutions by PDE systems

Hiroshi Ishii, Yoshitaro Tanaka

TL;DR

This work develops a PDE-based surrogate for nonlocal spatial convolutions with radial kernels in higher dimensions by expressing kernels as finite linear combinations of Green functions $k_j(x)=k(x;d_j)$ and approximating $K$ in the Sobolev space $H^m({\mathbb{R}}^n)$. It proves that for any radial kernel $K\in H^m({\mathbb{R}}^n)$ and error $\epsilon>0$, there exist $N$ and coefficients $\{\alpha_j\}$ such that $K_N=\sum_{j=1}^N \alpha_j k_j$ satisfies $\|K-K_N\|_{H^{m}}<\epsilon$, enabling $K*f$ to be approximated by $K_N*f$ in various norms. The completeness of the linear span of Green functions is established via an orthonormal basis in the radial Sobolev space $H^{m}_r({\mathbb{R}}^n)$ and analytic-function arguments, ensuring that every radial kernel can be represented by a finite Green-function sum. Numerical examples illustrate the practical construction of coefficients and the approximation behavior across dimensions, highlighting both the potential and open challenges (e.g., coefficient magnitudes and L^1-type results). Overall, the paper extends 1D PDE-approximation techniques to higher dimensions, enabling PDE-based treatment of nonlocal spatial convolutions in broader applications.

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.

On the approximation of spatial convolutions by PDE systems

TL;DR

This work develops a PDE-based surrogate for nonlocal spatial convolutions with radial kernels in higher dimensions by expressing kernels as finite linear combinations of Green functions and approximating in the Sobolev space . It proves that for any radial kernel and error , there exist and coefficients such that satisfies , enabling to be approximated by in various norms. The completeness of the linear span of Green functions is established via an orthonormal basis in the radial Sobolev space and analytic-function arguments, ensuring that every radial kernel can be represented by a finite Green-function sum. Numerical examples illustrate the practical construction of coefficients and the approximation behavior across dimensions, highlighting both the potential and open challenges (e.g., coefficient magnitudes and L^1-type results). Overall, the paper extends 1D PDE-approximation techniques to higher dimensions, enabling PDE-based treatment of nonlocal spatial convolutions in broader applications.

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.
Paper Structure (10 sections, 10 theorems, 60 equations, 1 figure)

This paper contains 10 sections, 10 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Approximation theorem
  3. Completeness of a linear span of the Green functions
  4. Properties of the Green function
  5. Linear span of the Green functions
  6. Completeness of the orthonormal basis
  7. Approximation of radial integral kernels
  8. Approximation of spatial convolutions
  9. Numerical example of the approximation of an integral kernel
  10. Concluding remarks

Key Result

Theorem 2.1

Let $m$ be a non-negative integer and $\{d_j\}_{j\in{\mathbb{N}}}$ be a sequence of distinct positive constants that has a positive accumulation point. Assume that $K\in H^{m}({\mathbb{R}}^n)$ is a radial function. Then, for any $\epsilon>0$, there exist $N\in{\mathbb{N}}$ greater than $(2m+n-1)/4$ and hold.

Figures (1)

  • Figure 1: Graphs of $K(x)$ and $K_N(x)$ with $N=10$ (top panels) and the distributions of $\{\alpha_j\}_{1\le j\le N+1}$ (bottom panels). The plots of $K(x)$ and $K_N(x)$ are presented by the red curve and blue-dashed curve, respectively. The numerical examples are presented for $n=1$ (left), $n=2$ (middle), and $n=3$ (right).

Theorems & Definitions (18)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • ...and 8 more