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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 reaction–diffusion) equations on smooth, closed surfaces embedded in \(\mathbb {R}^3\). 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.

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Acknowledgements

This work has been supported by Spanish MICINN Grant FIS2016-77892-R. We thank the anonymous reviewer for his or her careful reading of our manuscript and his or her many insightful comments and suggestions.

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Correspondence to Diego Álvarez.

Appendix A: Laplace–Beltrami Operator of an RBF on the Unit Sphere

Appendix A: Laplace–Beltrami Operator of an RBF on the Unit Sphere

We show that the expression for the Laplace–Beltrami operator of an RBF defined on a general surface (19) reduces to (20), when the given nodes \(\{\mathbf {x}_i\}\), \(i=1,\ldots ,N\), are located on the unit sphere.

Let \(\varPsi (\mathbf {x})=x^2+y^2+z^2-1\) be the function whose zero level set defines the unit sphere

$$\begin{aligned} \mathbb {S}^2=\{ \mathbf {x}\in \mathcal {R}^3\,:\,\,\varPsi (\mathbf {x})=0 \} \end{aligned}$$
(50)

implicitly. Then, \(\varvec{\nabla } \varPsi (\mathbf {x})=2\mathbf {x}\), so using (25) we obtain that \(\mathbf {n}(\mathbf {x})=\mathbf {x}\). Similarly, we find that \(\varDelta \varPsi (\mathbf {x})=6\) and \(\text {D}_{\mathbf {n}}\left( \varvec{\nabla } \varPsi (\mathbf {x})\right) = (\mathbf {x}\cdot \varvec{\nabla })\,2\mathbf {x}=2\mathbf {x}\), so the curvature is just \(\kappa (\mathbf {x})=2\). Then, using (19) it follows that

$$\begin{aligned} \varDelta _{\mathbb {S}^2}\phi (r_i(\mathbf {x}))= \left( 1+\frac{(\mathbf {x}\cdot \mathbf{{r}}_i)^2}{r_i^2} -2\,\mathbf {x}\cdot \mathbf{{r}}_i\right) \frac{1}{r_{i}}\displaystyle \frac{\text {d}\phi (r_i)}{\text {d}r_i} + \left( 1- \frac{(\mathbf {x}\cdot \mathbf{{r}_i})^2}{r_i^2}\right) \displaystyle \frac{\text {d}^2\phi (r_i)}{\text {d}r_i^2}. \end{aligned}$$
(51)

Since \(\mathbf {x}_i, \mathbf {x}\in \mathbb {S}^2\), \(||\mathbf {x}_i||=||\mathbf {x}||=1\), so from the law of cosines \(||\mathbf {x}_i||^2=||\mathbf{{r}}_i-\mathbf {x}||^2=r_{i}^2+||\mathbf {x}||^2-2r_{i}||\mathbf {x}||\cos (\mathbf{{r}}_i,\mathbf {x})\) we find that \(r_{i}=2\cos (\mathbf{{r}}_i,\mathbf {x})\) and, hence, \({\mathbf {x}\cdot \mathbf{{r}_i}}={||\mathbf {x}||r_i \cos (\mathbf{{r}}_i,\mathbf {x})}=r_i^2/2\). Using this result in (51) we finally obtain (20).

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Álvarez, D., González-Rodríguez, P. & Moscoso, M. A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator. J Sci Comput 77, 1115–1132 (2018). https://doi.org/10.1007/s10915-018-0739-1

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Keywords

  • Radial basis functions
  • Surface Laplacian
  • Surface PDE
  • Cardiology

Mathematics Subject Classification

  • 65D05
  • 35R01
  • 58J05