Given only a collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to solve elliptic partial differential equations (PDEs) supplemented with boundary conditions. Notice that 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 such constructions. 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 weak (variational) sense. The latter reduces the smoothness requirements on the underlying functions which is crucial to approximating weak solutions to PDEs. As a by-product, we also provide a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problems. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown and must be estimated from data) in order to correct the boundary error term in the diffusion maps construction. Finally, using this estimated distance, we illustrate how to impose Dirichlet, Neumann, and mixed boundary conditions for some common PDEs based on the Laplacian. Several numerical examples confirm our theoretical findings.
翻译:仅从嵌入欧几里德空间的里曼尼方块中抽取了一些抽取的点数,在此文件中,我们建议了一种新方法,解决以边界条件补充的极离部分差异方程式(PDEs),指出在未知方块上建造三角定位可能既困难又昂贵,在计算和数据要求方面,我们的目标是在不进行这种构建的情况下解决这些问题。相反,我们仅仅依靠抽样点来定义未知方块上的二次公式。我们的主要工具是扩散地图算法。我们从弱(变异)的角度重新分析这一众所周知的方法。后者降低了对接近PDEs的薄弱解决方案至关重要的基本功能的顺畅要求。作为一个副产品,我们还为扩散地图和Neumannegenvalu问题之间众所周知的关系提供了严格的理由。我们随后采用了一种最近开发的估算距离到边界函数的距离的方法(尤其是,边界位置被假定为未知,必须从数据中估算出来)。 后者降低了对PDiDides基本功能的光度要求,从而纠正对PDirequetal mindal imal deal eximations, eximation the the folviews the foltial demogradustration delistrational demogradustration.