Convergence, Stability, and Discrete Approximation of Laplace Spectra

Spectral methods have been widely used in a broad range of application fields. One important object involved in such methods is the Laplace-Beltrami operator of a manifold. Indeed, a variety of work in graphics and geometric optimization uses the eigen-structures (i.e, the eigenvalues and eigenfunctions) of the Laplace operator, and applications include mesh smoothing, compression, editing, shape segmentation, matching, and parameterization, among others. While the Laplace operator is defined (mathematically) for a smooth domain, in these applications, the underlying manifold is often approximated by a discrete mesh, and the spectral structure of the manifold Laplacian is estimated from some discrete Laplace operator constructed from this mesh. In this paper, we study the important question of how well the spectrum computed from the discrete mesh approximates the true spectrum of the manifold Laplacian. We exploit a recent result on mesh Laplacian and provide the first convergence res...

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Tamal K. Dey, Pawas Ranjan, Yusu Wang
• Keywords: Discrete Algorithms, Discrete Laplace Operator, Discrete Mesh, Manifold Laplacian, SODA 2010