An inverse theorem for compact Lipschitz regions in ℝ^{𝕕} using localized kernel bases

Abstract

While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the Lp norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction recently presented in [8].where in defining ρ(X, D) we assume that q(X) > 0.When there is no chance of confusion, we drop dependence in these parameters on X and D (referring simply to h, q and ρ).Remark 2.1. A finite fill distance h guarantees that the set D is covered by the family of balls B(ξ, h) := {x ∈ D | dist(x, ξ) < h}, ξ ∈ X. A positive separation radius q guarantees that B(ξ, q) ∩ B(ζ, q) = {} for distinct ζ, ξ ∈ Ξ. The mesh ratio, which automatically satisfies ρ ≥ 1, measures the uniformity of the distribution of X in D. The larger ρ(X, D) is, the less uniform the distribution is. If ρ is finite, then we say that the point set X is quasi-uniformly distributed (in D), or simply that X is quasi-uniform.Note that, for a compact subset D and a nonempty, finite subset X ⊂ D, the fill distance and separation radius are both positive and finite 0 < q < h < ∞. Consequently, ρ is finite, too.Many of the results in this article depend in some way on the geometry of the point set X -often this emerges in an estimate, where a constant depends on ρ. In most cases, (as one may expect) the strength of the estimate degrades as ρ increases. Throughout the paper, we have attempted make this control explicit, by factoring, whenever possible, the constant into a part which is totally independent of the point set, and another, which is a function of ρ.

Error Analysis of Nodal Meshless Methods

“…The optimal stencils decay extremely quickly away from the origin. This is predicted by results of [19] concerning exponential decay of Lagrangians of polyharmonic kernels, as used successfully in [13] to derive local inverse estimates. See [24] for an early reference on polyharmonic near-Lagrange functions.…”

$H^2$-Convergence of Least-Squares Kernel Collocation Methods

“…We use the notation u β,Z,Φm := zj ∈Z β j Φ m (· − z j ) to denote the functions in the trial space U Z,Φm spanned by translates of the kernel Φ m on the trial centers in Z with coefficients forming a vector β ∈ R |Z| . Then (I − ∆) ν u β,Z,Φm = u β,Z,Ψm−ν holds, and these are the functions that we use in [13,Eqn. 3.19].…”

A meshless Galerkin method for non-local diffusion using localized kernel bases

“…Meshless methods also allow for flexibility in the selection of approximating functions, in particular non-polynomial approximating functions. In this paper the approximating spaces will be spanned by certain localized kernel bases [10,13] that are distinguished by a rigorous approximation theory and give rise to very practical and efficient numerical methods.…”

“…The numerical analysis provided in this paper will be based on two specific classes of local Lagrange functions that will play the role of bases for the spaces U h and Λ h appearing in (2.6). In [13], it was shown that for either thin-plate splines or Matérn kernels on R n , local Lagrange functions with each function determined by O(log N ) n points contained in a ball of radius Kh log h centered at a given point ξ have very rapid decay around ξ. Moreover such functions generate very stable bases.…”

“…The theoretical development for such functions first appeared in [10] in the context of S 2 . The corresponding theory for compact domains in R n appeared in [13]. Applications using these basis functions in the context of numerical solution of certain PDEs have been given in [4,14,15].…”