Given only a finite collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to numerically solve elliptic and parabolic partial differential equations (PDEs) supplemented with boundary conditions. Since the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data requirements, our goal is to solve these problems without a triangulation. Instead, we rely only on using the sample points to define quadrature formulas on the unknown manifold. Our main tool is the diffusion maps algorithm. We re-analyze this well-known method in a variational sense for manifolds with boundary. Our main result is that the variational diffusion maps graph Laplacian is a consistent estimator of the Dirichlet energy on the manifold. This improves upon previous results and provides a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problem. Moreover, using semigeodesic coordinates we derive the first uniform asymptotic expansion of the diffusion maps kernel integral operator for manifolds with boundary. This expansion relies on a novel lemma which relates the extrinsic Euclidean distance to the coordinate norm in a normal collar of the boundary. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown) to construct a consistent estimator for boundary integrals. Finally, by combining these various estimators, we illustrate how to impose Dirichlet and Neumann conditions for some common PDEs based on the Laplacian. Several numerical examples illustrate our theoretical findings.

给定从嵌入欧几里德空间中的黎曼流形采样的有限点集合，在本文中，我们提出了一种新方法来数值求解补充了边界条件的椭圆和抛物型偏微分方程（PDE）。由于在未知流形上构建三角剖分在计算和数据要求方面都可能既困难又昂贵，因此我们的目标是在没有三角剖分的情况下解决这些问题。相反，我们仅依靠使用样本点来定义未知流形上的求积公式。我们的主要工具是扩散图算法。我们从变分意义上重新分析了这种众所周知的方法，适用于有边界的流形。我们的主要结果是，变分扩散图拉普拉斯算子是流形上狄利克雷能量的一致估计量。这改进了之前的结果，并为扩散图和诺依曼特征值问题之间众所周知的关系提供了严格的证明。此外，使用半测地坐标，我们推导了具有边界的流形的扩散映射核积分算子的第一个一致渐近展开。这种扩展依赖于一个新颖的引理，该引理将外在欧几里德距离与边界法线环中的坐标范数联系起来。然后，我们使用最近开发的估计边界函数距离的方法（请注意，假设边界位置未知）来构建边界积分的一致估计器。最后，通过结合这些不同的估计量，我们说明了如何对基于拉普拉斯算子的一​​些常见偏微分方程施加狄利克雷和诺依曼条件。