Approximation of the spectrum of a manifold by discretization

2013/01/16 by Erwann Aubry, Aubry, Erwann
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary value problem #Computer science #Curvature #Differential Geometry (math.DG) #Discretization #Eigenfunction #Eigenvalues and eigenvectors #FOS: Mathematics #Geometry #Laplace–Beltrami operator #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum mechanics #RADIUS #Ricci curvature #Riemannian manifold #Scalar curvature #Sectional curvature #Sequence (biology) #Spectral Theory (math.SP) #Spectral geometry #Spectral radius #Spectrum (functional analysis) #Topological and Geometric Data Analysis #Upper and lower bounds #advanced mathematical theories #math.AP #math.DG #math.SP #p-Laplacian

paper · pdf · doi:10.48550/arxiv.1301.3663

arxiv created 2013/01/16 · openalex publication_date 2013/01/16 · arxiv updated 2013/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sectional curvature and on a lower bound on the injectivity radius.

Citations

• Eigenvalue comparison theorems and its geometric applications

Related

• Graph approximations to the Laplacian spectra
• Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature
• A graph discretization of the Laplace-Beltrami operator
• A note on eigenvalue bounds for non-compact manifolds
• Eigenvalues Estimates for the p-Laplace Operator on Manifolds