Approximation of the spectrum of a manifold by discretization

TL;DR

This paper presents a method to approximate the spectral data of a compact Riemannian manifold using discrete Laplace operators on graphs, providing error bounds based on geometric properties.

Contribution

It introduces a new approach to discretize manifolds for spectral approximation with explicit error estimates depending on geometric bounds.

Findings

Error bounds depend on diameter, curvature, and injectivity radius.

Spectral data of graphs converges to that of the manifold.

Method is computable and applicable to various geometric settings.

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.