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Exploring Oversampling in RBF Least-Squares Collocation Method of Lines for Surface Diffusion

Meng Chen, Leevan Ling

TL;DR

This work addresses stable spatial discretization for surface diffusion PDEs using RBF-LSC-MoL by analyzing oversampling effects on the induced ODEs. The authors implement an oversampling strategy with ratio $n_X/n_Z$ and perform numerical experiments on isotropic and anisotropic surface diffusion on manifolds, using fully discretized schemes and a QR-based ODE reduction. They find that high kernel smoothness $m$ and larger center counts $n_Z$ can cause eigenvalue instability without oversampling, while modest oversampling ($1.5$–$2$) restores stability and efficiency. The results lead to practical guidelines for configuring RBF-LSC-MoL: ensure sufficient spatial approximation power with $n_Z$, and oversample by about $1.5$–$2$ to balance accuracy and computational cost, with a caveat about spurious zero eigenvalues due to rank deficiency. The work advances mesh-free methods for parabolic PDEs on surfaces by providing concrete, implementable recommendations.

Abstract

This paper investigates the numerical behavior of the radial basis functions least-squares collocation (RBF-LSC) method of lines (MoL) for solving surface diffusion problems, building upon the theoretical analysis presented in [SIAM J. Numer. Anal., 61 (3), 1386-1404}]. Specifically, we examine the impact of the oversampling ratio, defined as the number of collocation points used over the number of RBF centers for quasi-uniform sets, on the stability of the eigenvalues, time stepping sizes taken by Runge-Kutta methods, and overall accuracy of the method. By providing numerical evidence and insights, we demonstrate the importance of the oversampling ratio for achieving accurate and efficient solutions with the RBF-LSC-MoL method. Our results reveal that the oversampling ratio plays a critical role in determining the stability of the eigenvalues, and we provide guidelines for selecting an optimal oversampling ratio that balances accuracy and computational efficiency.

Exploring Oversampling in RBF Least-Squares Collocation Method of Lines for Surface Diffusion

TL;DR

This work addresses stable spatial discretization for surface diffusion PDEs using RBF-LSC-MoL by analyzing oversampling effects on the induced ODEs. The authors implement an oversampling strategy with ratio and perform numerical experiments on isotropic and anisotropic surface diffusion on manifolds, using fully discretized schemes and a QR-based ODE reduction. They find that high kernel smoothness and larger center counts can cause eigenvalue instability without oversampling, while modest oversampling (–) restores stability and efficiency. The results lead to practical guidelines for configuring RBF-LSC-MoL: ensure sufficient spatial approximation power with , and oversample by about – to balance accuracy and computational cost, with a caveat about spurious zero eigenvalues due to rank deficiency. The work advances mesh-free methods for parabolic PDEs on surfaces by providing concrete, implementable recommendations.

Abstract

This paper investigates the numerical behavior of the radial basis functions least-squares collocation (RBF-LSC) method of lines (MoL) for solving surface diffusion problems, building upon the theoretical analysis presented in [SIAM J. Numer. Anal., 61 (3), 1386-1404}]. Specifically, we examine the impact of the oversampling ratio, defined as the number of collocation points used over the number of RBF centers for quasi-uniform sets, on the stability of the eigenvalues, time stepping sizes taken by Runge-Kutta methods, and overall accuracy of the method. By providing numerical evidence and insights, we demonstrate the importance of the oversampling ratio for achieving accurate and efficient solutions with the RBF-LSC-MoL method. Our results reveal that the oversampling ratio plays a critical role in determining the stability of the eigenvalues, and we provide guidelines for selecting an optimal oversampling ratio that balances accuracy and computational efficiency.
Paper Structure (4 sections, 24 equations, 13 figures, 3 tables)

This paper contains 4 sections, 24 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Surface diffusion: notations and preliminaries
  3. RBF-collocation method of lines
  4. Oversampling vs Eigenvalue stability

Figures (13)

  • Figure 1: Exmp. \ref{['ex1']}--\ref{['ex2']}: Schematic demonstration of data point distributions. (Black) Quasi-uniform data points on the unit sphere, and (Red) Regular grid in some narrow band domain containing the unit sphere.
  • Figure 2: Exmp. \ref{['ex1']}: We observe stable eigenvalues in $\mathbb{C}$ when both the kernel order of smoothness $m$ and the number of RBF centers $n_Z$ are relative small; oversampling does not have any obvious effect in terms of eigenvalue distribution. The provided legend applies to both subfigures: (TOP) full spectra, and (BOTTOM) Zoom-in near the origin.
  • Figure 3: Exmp. \ref{['ex1']}: RBF method of lines without any oversampling has unstable eigenvalues when either $m$ and/or $n_Z$ increases.
  • Figure 4: Exmp. \ref{['ex1']}: By imposing $1.5\times$ oversampling ratio, we consistently observe stable eigenvalues in $\mathbb{C}$ when using kernel with high order of smoothness and more RBF centers. Do note the spurious zero eigenvalue. Subfigures for $m=6$, $8$, and $10$ here are presented in same style as Figure \ref{['fig1']}.
  • Figure 5: Exmp. \ref{['ex1']}: Summaries of ODE45 results. (TOP) Current time against the number of adaptive time steps, (BOTTOM) maximum errors of various RBF method of lines solutions against time.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4