A closed-form formula for the RBF-based approximation of the Laplace-Beltrami operator
Radial basis function interpolation and quasi-interpolation provide a useful approach to the numerical solution of partial differential equations. The reasons why they are excellent candidates for improving classical methods such as finite elements (whose disadvantages include the need for expensive recomputation when nodes are added) and parametrisations of manifolds into flat surfaces (which can lead to singularities, e.g., of the north and south poles when the PDEs are related to spheres) are multiple: radial basis functions are available for an arbitrary number of scattered data in any dimension, they give high, sometimes spectral, asymptotic convergence, they can be computed in many cases efficiently and stably, and they are flexible using adaptive parameters such as the \(c\) in the most often used multiquadrics RBF \(\phi(r)=\sqrt{r^2+c^2}\). Sometimes local RBF methods are used which are similar to finite difference methods (sometimes in fact called RBF-FD algorithms), in that differences of shifts of radial basis functions that decay are used as bases instead. These are called also quasi-Lagrange functions in view of the Lagrange functions for interpolants. In the case of the asymptotically increasing multiquadrics for instance, these quasi-Lagrange functions decay at an algebraic (negative) rate. In this article, Laplace Beltrami operators are used to define the equations (e.g., diffusion, reaction-diffusion) to be solved on smooth surfaces in three dimensions. The paper provides closed forms of the operator and of the radial basis function approximation. The approximation is represented implicitly which has the advantage that the surface never has to be computed explicitly, it is represented by level sets, and the approximation is a function from which normals and curvature are computed. The implicit function \(\Psi\) is defined such that it is zero on the manifold and does not vanish at points outside. In the implementation, these outside points are chosen symmetrically on both sides of the surface, and the radial basis function interpolant \(s\) is defined by zero value at the data points on the manifold and nonzero outside, with function values that are proportional to the distance of the centre from the manifold (the distance related to the normal vectors). The closed form of the Laplace Beltrami operator is exact on the radial basis functions which is a suitable property to exploit the accuracy of the RBF approximation itself. In other words, no accuracy is lost in the numerical computation by the radial basis function approximation itself. Applications include pattern formations on growing biological surfaces, ice accreditation and flow on cold surfaces (airplanes), mantle flows in geosciences, and restoring damaged surface patterns. Here, an example from cardiology is given and its accuracy studied.
scientific article; zbMATH DE number 6993318
scientific article