Modified Virtual Grid Difference for Discretizing the Laplace--Beltrami Operator on Point Clouds

Meng Wang Shing-Yu Leung Hongkai Zhao

Numerical Analysis and Scientific Computing mathscidoc:1912.43192

SIAM Journal on Scientific Computing, 40, (1), A1-A21, 2018.1
We propose a new and simple discretization, named the modified virtual grid difference (MVGD), for numerical approximation of the Laplace--Beltrami (LB) operator on manifolds sampled by point clouds. The key observation is that both the manifold and a function defined on it can both be parametrized in a local Cartesian coordinate system and approximated using least squares. Based on the above observation, we first introduce a local virtual grid with a scale adapted to the sampling density centered at each point. Then we propose a modified finite difference scheme on the virtual grid to discretize the LB operator. Instead of using the local least squares values on all virtual grid points like the typical finite difference method, we use the function value explicitly at the grid located at the center (coinciding with the data point). The new discretization provides more diagonal dominance to the resulting linear system and
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@inproceedings{meng2018modified,
  title={Modified Virtual Grid Difference for Discretizing the Laplace--Beltrami Operator on Point Clouds},
  author={Meng Wang, Shing-Yu Leung, and Hongkai Zhao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112842777938752},
  booktitle={SIAM Journal on Scientific Computing},
  volume={40},
  number={1},
  pages={A1-A21},
  year={2018},
}
Meng Wang, Shing-Yu Leung, and Hongkai Zhao. Modified Virtual Grid Difference for Discretizing the Laplace--Beltrami Operator on Point Clouds. 2018. Vol. 40. In SIAM Journal on Scientific Computing. pp.A1-A21. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112842777938752.
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