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

A Closed-Form Formula for the RBF-Based Approximation of the Laplace–Beltrami Operator 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 $$\mathbb {R}^3$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
DOI
10.1007/s10915-018-0739-1
Publisher site
See Article on Publisher Site

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 $$\mathbb {R}^3$$ 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.

Journal

Journal of Scientific Computing – Springer Journals

Published: May 29, 2018

References

  • Shape reconstruction of cardiac ischemia from non-contact intracardiac recordings: a model study
    Alvarez, D; Alonso-Atienza, F; Rojo-Alvarez, JL; Garcia-Alberola, A; Moscoso, M
  • Coupling field theory with continuum mechanics: a simulation of domain formation in giant unilamellar vesicles
    Ayton, GS; McWhirter, JL; McMurty, P; Voth, GA
  • The surface finite element method for pattern formation on evolving biological surfaces
    Barreira, R; Elliott, CM; Madzvamuse, A
  • RBF-FD formulas and convergence properties
    Bayona, V; Moscoso, M; Carretero, M; Kindelan, M
  • Radial Basis Functions
    Buhmann, M
  • Inverse Problem of Electrocardiography: Estimating the Location of Cardiac Ischemia in a 3D Realistic Geometry
    Chavez, C; Zemzemi, N; Coudiere, Y; Alonso-Atienza, F; Alvarez, D
  • Interpolation in the limit of increasingly flat radial basis functions
    Driscoll, TA; Fornberg, B
  • Finite element methods for surface PDEs
    Dziuk, G; Elliott, CM
  • Advances in Multiresolution for Geometric Modelling
    Floater, M; Hormann, K
  • Transport schemes on a sphere using radial basis functions
    Flyer, N; Wright, GB
  • A radial basis function method for the shallow water equations on a sphere
    Flyer, N; Wright, GB
  • A stable algorithm for flat radial basis functions on a sphere
    Fornberg, B; Piret, C
  • Scattered data on embedded submanifolds with restricted positive definite kernel: Sobolev error estimates
    Fuselier, EJ; Wright, GB
  • A high-order kernel method for diffusion and reaction–diffusion equations on surfaces
    Fuselier, EJ; Wright, GB
  • Laurent series based RBF-FD method to avoid ill-conditioning
    González-Rodríguez, P; Bayona, V-; Moscoso, M; Kindelan, M
  • An improvement of a recent Eulerian method for solving PDEs on general geometries
    Greer, JB
  • A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant
    James, AJ; Lowengrub, J
  • Multiquadrics, a scattered data approximation scheme with applications to computational fluid dynamics. I. Surface approximations and partial derivatives estimates
    Kansa, EJ
  • Multiquadrics, a scattered data approximation scheme with applications to computational fluid dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations
    Kansa, EJ
  • Radial basis function interpolation in the limit of increasingly flat basis functions
    Kindelan, M; Moscoso, M; González-Rodríguez, P
  • The implicit closest point method for the numerical solution of partial differential equations on surfaces
    Macdonald, CB; Ruuth, SJ
  • Implicit brain imaging
    Memoli, F; Sapiro, G; Thompson, P
  • A two-current model for the dynamics of cardiac membrane
    Mitchell, CC; Shaeffer, DG
  • Interpolation of scattered data: distance matrices and conditionally positive definite functions
    Micchelli, CA
  • A mathematical model for atmospheric ice accretion and water flow on a cold surface
    Myers, TG; Charpin, JPF
  • The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces
    Piret, C
  • A simple embedding method for solving partial differential equations on surfaces
    Ruuth, SJ; Merriman, B
  • Multivariate interpolation by polynomials and radial basis functions
    Schaback, R
  • A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces
    Shankar, V; Wright, GB; Kirby, RM; Fogelson, AL
  • Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations
    Shu, C; Ding, H; Yeo, KS
  • Flows on surfaces of arbitrary topology
    Stam, J
  • Phase separation patterns for diblock copolymers on spherical surfaces: a finite volume method
    Tang, P; Qiu, F; Zhang, H; Yang, Y
  • On using radial basis functions in a “finite difference mode” with applications to elasticity problems
    Tolstykh, AI; Shirobokov, DA
  • A hybrid radial basis function-pseudospectral method for thermal convection in a 3D spherical shell
    Wright, GB; Flyer, N; Yuen, DA
  • How good can polynomial interpolation on the sphere be?
    Womersley, RS; Sloan, I

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