Generalized Moving Least-Squares Methods for Solving Vector-valued PDEs on Unknown Manifolds

TL;DR

This work develops two mesh-free GMLS-based schemes, intrinsic and extrinsic, to solve vector-valued PDEs on unknown smooth manifolds embedded in high-dimensional spaces from random point clouds. The intrinsic approach uses Monge parametrization to simplify geometric quantities and assemble a sparse $dN\times dN$ Bochner Laplacian and covariant-derivative operators, while the extrinsic approach projects ambient derivatives and employs a dimension-reduction $(n\to d)$ to produce a sparse $dN\times dN$ discretization. The authors provide rigorous formulations, complexity analyses, and extensive numerical validations across eigenvalue problems, Poisson-type equations, vector diffusion, and nonlinear Burgers’ equations on spheres and tori in varying ambient dimensions, showing optimal convergence rates and practical stability. The results demonstrate a scalable, mesh-free framework for vector PDEs on manifolds that is robust to unknown geometry and capable of handling high ambient dimensions, with potential extensions to boundary conditions and higher intrinsic dimensions.

Abstract

In this paper, we extend the Generalized Moving Least-Squares (GMLS) method in two different ways to solve the vector-valued PDEs on unknown smooth 2D manifolds without boundaries embedded in $\mathbb{R}^{3}$, identified with randomly sampled point cloud data. The two approaches are referred to as the intrinsic method and the extrinsic method. For the intrinsic method which relies on local approximations of metric tensors, we simplify the formula of Laplacians and covariant derivatives acting on vector fields at the base point by calculating them in a local Monge coordinate system. On the other hand, the extrinsic method formulates tangential derivatives on a submanifold as the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. One challenge of this method is that the discretization of vector Laplacians yields a matrix whose size relies on the ambient dimension. To overcome this issue, we reduce the dimension of vector Laplacian matrices by employing a coordinate transformation. The complexity of both methods scales well with the dimension of manifolds rather than the ambient dimension. We also present supporting numerical examples, including eigenvalue problems, linear Poisson equations, and nonlinear Burgers' equations, to examine the numerical accuracy of proposed methods on various smooth manifolds.

Figures (9)

• Figure 1: Sketch of our setups. (a) Global tangent-vector bases $\{\boldsymbol{t}_1,\boldsymbol{t}_2\}$ on the point cloud $\{\mathbf{x}_{i}\}_{i=1}^{N}$ (gray dots) without alignment. (b) $K$-nearest neighbors of the base point $\mathbf{x}_{0}$ plotted as blue crosses inside the red circle. (c) The tangent space $\mathrm{span}\{\boldsymbol{t}_1,\boldsymbol{t}_2\}$ and the normal $\boldsymbol{n}$ at the base point $\mathbf{x}_{0}$. Also plotted are their estimations $\{\boldsymbol{\hat{t}}_1,\boldsymbol{\hat{t}}_2, \boldsymbol{\hat{n}}\}$ given in (\ref{['eq:that']}) when the manifold is unknown. (d) The local Monge coordinate $(\theta_1,\theta_2)$ and the GMLS approximation of the manifold $\hat{q}(\theta_1,\theta_2)$ in the local coordinate system (equation (\ref{['eqn:qhat']})).
• Figure 2: Convergence results for the estimations of projection matrices $\hat{\mathbf{P}}$ on various manifolds. (a) 2D torus in $\mathbb{R}^3$. (b) 2D torus in $\mathbb{R}^9$. (c) 3D flat torus in $\mathbb{R}^{12}$. Here, $\Vert \cdot \Vert_{\mathrm{F}}$ denotes the Frobenius matrix norm, the projection matrix is ${\mathbf{P}}=\sum_{i=1}^d\boldsymbol{t}_{i}\boldsymbol{t}^\top_{i}$ and its estimation is $\hat{\mathbf{P}}=\sum_{i=1}^d\boldsymbol{\hat{t}}_{i}\boldsymbol{\hat{t}}^\top_{i}$ (see Section \ref{['sec:revext']} or harlim2023radial for more about the projection matrix).
• Figure 3: Eigenvalues on 2D sphere in $\mathbb{R}^3$. $K = 50$, $l = 5$, and $N = 6400$ points are randomly sampled. (Left) All eigenvalues lie in the left half complex plane. (Right) Comparison of the leading 8 eigenvalues of the Bochner Laplacian between our approximation using intrinsic and extrinsic GMLS methods and the analytic truth.
• Figure 4: Eigenvalues on 2D torus in $\mathbb{R}^3$. $K = 50$, $l = 5$, and $N = 6400$ points are randomly sampled. The first row: (a) Almost all eigenvalues lie in the left half complex plane. (Right) Comparison of the leading 8 eigenvalues of the Hodge Laplacian between the GMLS and the semi-analytic truth. The second row: (b) The first two eigenvalues of our approximation (with one positive and the other negative) converge to zero with $O(N^{-2})$. (c) and (d) are the two corresponding harmonic fields of our approximation.
• Figure 5: Screened Poisson problems on 2D sphere in $\mathbb{R}^3$ associated with the Hodge Laplacian. Random data with 6 trials. $K=50$ nearest neighbors are used. In panels (a) and (b), shown are the average IEs using extrinsic and intrinsic methods, respectively. In panels (c), shown are the IEs of the solutions using FEEC and RBF. In panels (d), shown are the comparison of the time cost among different methods. For time comparison, we use degree 2 for GMLS and FEEC.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Proposition 3.1
• Proposition 3.2
• proof