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We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
1922
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Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory
1956
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J. Dodziuk, V. K. Patodi, Riemannian structures and triangulations of manifolds
1977
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J. Cheeger, A vanishing theorem for piecewise constant curvature spaces
1986
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M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds
1990
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P. Berard, G. Besson, S. Gallot, Embedding Riemannian manifolds by their heat kernel
1994
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K. Fujiwara, Eigenvalues of Laplacians on a closed Riemannian manifold and its nets
1995
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Ph. G. Ciarlet, The finite element method for elliptic problems
2002
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I. Babuška, U, Banerjee, J. Osborn, Survey of meshless and generalized finite element methods: a unified approach
2003
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A. Petrunin, Polyhedral approximations of Riemannian manifolds
2003
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M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem
2004
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M. Belkin, P. Niyogi, Convergence of Laplacian Eigenmaps
2007
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A. Katsuda, Y. Kurylev and M. Lassas, Stability of boundary distance representation and reconstruction of Riemannian manifolds
2007
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D. Panov, Polyhedral Kähler manifolds
2009
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B. Poonen, Undecidable problems: a sampler
2012
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E. Aubry, Approximation of the spectrum of a manifold by discretization
2013
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