Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

Abstract

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ∞ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L 2 minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

“…Both (7) and (6) (7).) Similar bounds hold for Lagrange functions associated with other kernels, both positive definite and conditionally positive definite, as discussed in [15] and [12].…”

“…Similar results hold for other kernels on specific compact manifolds [15]. In the case where the manifold is not compact, one typically is more interested in Lagrange functions based on finite point sets which are quasiuniform with respect to a compact subset Ω ⊂ M. Nevertheless, a similar pointwise decay estimate for Lagrange functions holds for that setting as well [12,Inequality 3.5].…”

“…Using the triangle inequality, the bound in (15), and the estimate above, we have the following result: Lemma 6. Let M be as in Section 2.1 and let κ m be a Sobolev-Matérn kernel.…”

A novel Galerkin method for solving PDES on the sphere using highly localized kernel bases

“…Making use of this observation yields a fractional order "zeros lemma," similar to integer order ones proved in [13,Appendix A]. This will be important in the sequel.…”

Zooming from global to local: a multiscale RBF approach

“…Assume that a finite set X ⊂ Ω has a sufficiently small (local) mesh norm h X,Ω . Then, for any function u ∈ H σ (Ω), σ > d/2, with u| X = 0,for all 0 ≤ ν ≤ σ we have u H ν (Ω) ≤ Ch σ−ν X,Ω u H σ (Ω) .Proof: For Ω ⊂ S d an open and connected set with Lipschitz boundary, the proof follows from the zeros lemma for Lipschitz domains on a Riemannian manifold in[7, Theorem A.11]. The case Ω = S d was proved earlier in[10].…”