Multilevel interpolation of divergence-free vector fields

Abstract

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.

An analytically divergence-free collocation method for the incompressible Navier-Stokes equations on the rotating sphere

“…See also [21] for the method applied to the time-dependent Stokes equation. Kernel-based divergence-free approximation methods have been studied and analysed in [7,3,11,14,12,22,23,8], and were successfully applied to different problems, see for example, [27,28,34]. In [25] and [13], the authors extended and analysed the theory of divergence-free and rotation-free kernel-based approximations for tangential vector fields on manifolds, especially the unit sphere.…”

“…Using standard time discretisation methods such as Runge-Kutta methods, the system given in (7) can be turned into a fully discretised system. Hence, the linear system to determine β j for (8) as well as the linear system to determine the α j in (6) have to be solved in every time-step. However, choosing kernel functions Φ with compact support, the matrices in both linear systems can be made sparse, such that both linear systems can be solved sufficiently fast.…”

“…Finally, it leads easily to a high-order approximation of the pressure just by exchanging the kernel function in (8), i.e.…”

Interpolation with uncoupled separable matrix-valued kernels

“…For divergence-free kernels, matrix-valued kernel approximations have already been succesfully applied, see e.g. ( [7,13,20,9]). This paper is structured as follows: In Section 2 we begin with an introduction to matrix-valued kernels and extend well-known properties from the scalar-valued case including error estimation.…”

Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Matérn kernels

“…Hence, a natural application of such profiles ψ may occur in the field of multilevel interpolation algorithms; cf. Farrell et al [12]. Another potential application, to tomographic image reconstruction via X-ray transform, is discussed by Ward and Unser [23], where the plots of the profiles η 2 and ψ 3,2 , rescaled on the support interval [0, 1], are also presented.…”