Diffusion Maps for Embedded Manifolds with Boundary with Applications to PDEs-猫眼课题宝

DOI：

10.1016/j.acha.2023.101593

发表时间：

2019-11

期刊：

ArXiv

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

Ryan Vaughn;Tyrus Berry;Harbir Antil

给出了欧氏空间中黎曼流形上采样点的有限集合，提出了一种新的方法来数值求解带有边界条件的椭圆型和抛物型偏微分方程组。由于在未知流形上构建三角剖分可能既困难又昂贵，无论是在计算还是数据要求方面，我们的目标是在不进行三角剖分的情况下解决这些问题。相反，我们仅依靠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。对于带边界的流形，我们用变分的方法重新分析了这一著名的方法。我们的主要结果是变分扩散映射图的拉普拉斯是流形上狄利克雷能量的一致估计。这改进了以前的结果，并为扩散映射和Neumann特征值问题之间的众所周知的关系提供了严格的证明。此外，利用半偶坐标，我们得到了带边界流形的扩散映射核积分算子的第一一致渐近展开式。这一展开依赖于一个新的引理，该引理将外欧几里德距离与边界的法环中的坐标范数联系起来。然后，我们使用最近发展的估计到边界函数的距离的方法(注意，边界位置被假设为未知)来构造边界积分的一致估计器。最后，通过组合这些不同的估计量，我们说明了如何基于拉普拉斯对一些常见的偏微分方程加Dirichlet和Neumann条件。几个数值例子说明了我们的理论发现。

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.

Diffusion Maps for Embedded Manifolds with Boundary with Applications to PDEs

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