Solving eigenvalue problems on curved surfaces using the Closest Point Method

Abstract

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. or, more generally, the elliptic operatorThe Laplace-Beltrami eigenvalue problem has played a prominent role in recent years in data analysis. For example, in [1], eigenvalues of the Laplace-Beltrami operator were used to extract "fingerprints" which characterize surfaces and solid objects. In [2, 3], Laplace-Beltrami eigenvalues and eigenfunctions were used for dimensionality reduction and data representation. Other application areas include smoothing of surfaces [4] and the segmentation and registration of shape [5].Analytical solutions to the Laplace-Beltrami eigenvalue problem are rarely available, so it is crucial to be able to numerically approximate them in an accurate and efficient manner. Partial differential equations on surfaces, including eigenvalue problems, have traditionally been approximated using either (a) discretizations based on a parameterization of the surface [6], (b) finite element discretizations on a triangulation of the surface [7], or (c) embedding techniques which solve some embedding PDE in a small region near the surface [8] (see also the related works [9,10,11,12,13,14,15]).Parameterization methods (a) are often effective for simple surfaces [6], but for more complicated geometries have the deficiency of introducing distortions and singularities into the method through the parameterization [16]. Approaches based on the finite element method can be deceptively difficult to implement; as described in [7], "even though this method seems to be very simple, it is quite tricky to implement". Embedding methods (c) have gained a considerable following because they permit PDEs on surfaces to be solving using standard finite differences.This paper proposes a simple and effective embedding method for the Laplace-Beltrami eigenvalue problem based on the Closest Point Method. The Closest Point Method is a recent embedding method that has been used to compute the numerical solution to a variety of partial differential equations [17,18,19,20], including in-surface heat flow, reaction-diffusion equations, and higher-order motions involving biharmonic and "surface diffusion" terms. Unlike traditional embedding methods, which are built around level set representatives of the surface, the Closest Point Method i...

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