Generalized Finite Difference Method on unknown manifolds

TL;DR

This paper extends the Generalized Finite Difference Method (GFDM) to unknown manifolds using sampled data, providing theoretical convergence guarantees and novel techniques for boundary detection and approximation of differential operators.

Contribution

We formalize GFDM on unknown manifolds with Taylor expansions, introduce a stable approximation method for the Laplace-Beltrami operator, and develop a boundary detection technique without prior boundary knowledge.

Findings

GFDM achieves stable approximation of Laplace-Beltrami operator.

Theoretical convergence of GFDM for Poisson PDEs is established.

Numerical experiments demonstrate high accuracy on various manifolds.

Abstract

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and require the knowledge of the boundary location.