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SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS
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DOI：10.1137/120869730.png)
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Abstract
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In this paper we present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parameterization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis using K nearest neighbors. Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.
Keywords:
manifold
gradient
Laplace-Beltrami operator
principle component analysis
moving least squares
constrained quadratic optimization
time dependent PDE
upwind scheme
semi-Lagrangian method
eigenvalue problem
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