A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator

Abstract

In this paper we present a method that uses radial basis functions to approximate the Laplace-Beltrami operator that allows to solve numerically diffusion (and reactiondiffusion) equations on smooth, closed surfaces embedded in R 3 . The novelty of the method is in a closed-form formula for the Laplace-Beltrami operator derived in the paper, which involve the normal vector and the curvature at a set of points on the surface of interest. An advantage of the proposed method is that it does not rely on the explicit knowledge of the surface, which can be simply defined by a set of scattered nodes. In that case, the surface is represented by a level set function from which we can compute the needed normal vectors and the curvature. The formula for the Laplace-Beltrami operator is exact for radial basis functions and it also depends on the first and second derivatives of these functions at the scattered nodes that define the surface. We analyze the converge of the method and we present numerical simulations that show its performance. We include an application that arises in cardiology.

A local radial basis function method for the Laplace-Beltrami operator

“…Then, we describe how to compute the weights for local RBF-FD approximation of the LBO. As in [2], it is also necessary to compute some characteristics of the surface, such as the normal vectors and its curvature, which are also computed using a local RBF methodology. Finally, we describe how to build a differentiation matrix that approximates the action of the LBO in reaction-diffusion type equations on surfaces.…”

“…There are several types of RBFs, and the choice of the optimal one for a given problem is still an open question. Among the infinitely differentiable RBFs, those used most often are the Gaussian φ(r) = exp(−( r) 2 ), the Inverse Quadratic φ(r) = 1/(1 + ( r) 2 ) and the Inverse Multiquadric φ(r) = 1/ 1 + ( r) 2 . All these contain a free parameter , called shape parameter, that controls the flatness of the RBF (the smaller the flatter).…”

“…In [2], we derived a closed-form formula for the (global) RBF-based approximation of the Laplace Beltrami Operator (LBO). It can not only be applied to surfaces whose explicit formula is known, but also to surfaces defined by a set of scattered nodes.…”

“…In this paper, we generalize the approach in [2] to local RBF approximations. We carry out a thorough study of the proposed method and analyze its convergence and stability properties when it is applied to reaction diffusion equations over surfaces.…”

Computing Eigenmodes of Elliptic Operators on Manifolds Using Radial Basis Functions

“…For example, see [2] for the closed-form formulations of orthogonal projection method. In addition, several recent articles presented hybrid methods that incorporate features of RBF approach alongside embedding methods [53].…”