Two-step Generalized RBF-Generated Finite Difference Method on Manifolds

TL;DR

The paper tackles solving PDEs on manifolds from randomly sampled point clouds by introducing a two-step generalized RBF-FD (gRBF-FD) method that first regresses with GMLS in tangent-space coordinates and then compensates the residual with a PHS interpolation. It leverages Monge parametrization to express operators in local coordinates, uses a weight-based stabilization to produce nearly diagonally dominant Laplacians, and introduces an automatic K-nearest neighbor tuning strategy to ensure numerical stability. The authors prove error bounds in terms of stencil diameter and data count, show an overall convergence rate of O((log N / N)^{(l-1)/d}) for random data, and demonstrate superior accuracy and stability of gRBF-FD over GMLS across a suite of challenging manifolds, including RBC, BSP, flat tori, and complex 3D surfaces. These advances yield a scalable, mesh-free framework for high-accuracy Laplace-Beltrami approximations on point clouds, with potential impact on physics-informed modeling, geometry processing, and surface PDEs.

Abstract

Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step generalized radial basis function-generated finite difference (gRBF-FD) method for solving PDEs on manifolds without boundaries, identified by randomly sampled point cloud data. The gRBF-FD is based on polyharmonic spline kernels and multivariate polynomials (PHS+Poly) defined over the tangent space in a local Monge coordinate system. The first step is to regress the local target function using a generalized moving least squares (GMLS) while the second step is to compensate for the residual using a PHS interpolation. Our gRBF-FD method has the same interpolant form with the standard RBF-FD but differs in interpolation coefficients. Our approach utilizes a specific weight function in both the GMLS and PHS steps and implements an automatic tuning strategy for the stencil size K (i.e., the number of nearest neighbors) at each point. These strategies are designed to produce a Laplacian matrix with a specific coefficient structure, thereby enhancing stability and reducing the solution error. We establish an error bound for the operator approximation in terms of the so-called local stencil diameter as well as in terms of the number of data. We further demonstrate the high accuracy of gRBF-FD through numerical tests on various smooth manifolds.

Figures (13)

• Figure 1: 1D ellipse in$\mathbb{R}^2$. Comparison among GMLS (with three different weights), RBF-FD and gRBF-FD. The upper panels are well-sampled data and the bottom panels are random data. The left(a)(d), middle(b)(e), right(c)(f) columns correspond to the scattered dataset, coefficient $w_k$ vs. $k$th nearest neighbor, $\Vert \boldsymbol{L}_{\mathbf{X}_{M},\boldsymbol{I}}^{-1}\Vert _{\infty }$ vs. $N$, respectively. We fix $K=30$, polynomial degree $4$ and PHS parameter $\kappa=3$.
• Figure 2: 1D ellipse in$\mathbb{R}^{2}$. Comparison among GMLS (with three weight functions), RBF-FD and gRBF-FD. The left panels are well-sampled data and the right panels are random data. The upper panels are forward error (FE) vs. $N$ while the bottom panels are inverse error (IE) vs. $N$. We fix $K=30$, polynomial degree $4$ and PHS parameter $\kappa =3$.
• Figure 3: 1D ellipse in$\mathbb{R}^{2}$. (Left) Stencil diameter as a function $\theta$ for $N=400$ points. Plotted are the sizes of all stencils for well-sampled data (middle, light blue circles) and random data (right, orange circles). Each circle is plotted with a center $\mathbf{x}_{i}$ and a radius $D_{K,\max }({\mathbf{x}_{i}})/2$. The circle plotted using dark blue (middle) and dark red (right) corresponds to the largest stencil diameter. We fix $K=30$, polynomial degree $4$ and PHS parameter $\kappa =3$.
• Figure 4: 1D ellipse in$\mathbb{R}^2$ using random data. Shown are (a) the Laplacian coefficient $w_k$ vs. $k$th nearest neighbor, (b) $\Vert\boldsymbol{L}^{-1}_{\mathbf{X}_M,\boldsymbol{I}}\Vert_{\infty}$ vs. $N$, and (c) the leading 200 eigenvalues of $\boldsymbol{L}_{\mathbf{X}_M}$ for different $K$ neighbors. We fix polynomial degree 4 and PHS parameter $\kappa=3$ for all panels. In panels (a)(c), we use $N=1600$ random data points.
• Figure 5: 1D ellipse in$\mathbb{R}^2$ using random data. Shown are (a) the coefficient of the center point $w_1$ vs. the stencil size $K$ and (b) the ratio $\gamma$ vs. $K$. In both panels (a) and (b), we show the results on three different stencils. We fix the number of points $N=1600$, polynomial degree 4 and PHS parameter $\kappa=3$.

Theorems & Definitions (17)

• Remark 3.1
• Remark 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Example 4.1
• Definition 4.2
• Definition 4.3
• Lemma 4.4
• Lemma 4.5