Geometric PDEs Based Methods for Analyzing Point Clouds in 3D and Higher

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Recent advances of sensor and data collection techniques have led to "data deluge" from wide applications in science and engineering. In order for data scientists to make sense of this massive amount of data, it becomes increasingly important to develop new tools in both theoretical and computational point of views for data representation, analysis and mining. In practice, it is natural to represent data as a collection of points sampled from a low dimensional manifold in a high dimension space. In this scenario, intrinsic geometric information extraction is a crucial step to conduct high level structure understanding of data.The PI will investigate geometry and partial differential equation (PDE) based methods for analyzing point clouds data in 3D and higher dimension. One objective is to develop and improve numerical methods for solving PDEs on point clouds sampled from Riemannian manifolds. Another one is to utilize solutions of specially designed geometric PDEs on point clouds to conduct intrinsic and global understanding of data sets, such as topological information extraction and point clouds intrinsic comparisons. This investigation will lead to different viewpoints for tackling problems in intrinsic data analysis. The research also contributes to related areas in numerical methods for solving PDEs, computational differential geometry and non-convex optimization. By collaborating with computer scientists and biomedical engineers, applications to real problems will also be explored by the PI.