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

TL;DR

This paper introduces a novel two-step gRBF-FD method for solving PDEs on manifolds from point cloud data, improving stability and accuracy through specialized interpolation and automatic stencil tuning.

Contribution

The paper develops a new two-step gRBF-FD approach with automatic stencil size tuning, enhancing stability and accuracy for PDEs on manifolds from point clouds.

Findings

High accuracy demonstrated on various smooth manifolds

Enhanced stability due to specific coefficient structure

Error bounds established for operator approximation

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.