Error bounds for kernel-based numerical differentiation

Abstract

The literature on meshless methods observed that kernel-based numerical differentiation formulae are robust and provide high accuracy at low cost. This paper analyzes the error of such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we have a convenient way to compare kernel-based and polynomial-based formulas. It follows that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. A variety of examples is provided.

A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs

“…The R 3 function we have used is given by 11) and is illustrated in Figure 2.2. For the theoretical convergence estimates derived in Section 4, we need a wellposedness estimate that relates the norm of the solution u of (2.1) to the data f and g. For the case f ≡ 0, problem (2.1) is reduced to the Laplace equation, and the maximum principle holds for the solution u g u g L∞(Ω) ≤ g L∞(∂Ω) .…”

Error Analysis of Nodal Meshless Methods

“…The generalized finite difference approximations may be calculated via radial kernels using local selections of nodes only [25,36,35], and there are papers on how to calculate such approximations, e.g. [7,18]. Bypassing Moving Least Squares trial functions, direct methods in the context of Meshless Local Petrov Galerkin techniques are in [21,20], connected to diffuse derivatives [23].…”

“…In [7], these problems were partly overcome by variable precision arithmetic, while the paper [18] provides a very nice stabilization technique, but unfortunately confined to approximations based on the Gaussian kernel. We hope to be able to deal with stabilization of the evaluation of the quadratic form in a forthcoming paper.…”

“…[21,7] in terms of u * S , with explicit convergence orders in terms of powers We call there orders consistency orders in what follows. Except for Section 6, we do not survey such results here, but users can be sure that a sufficiently fine fill distance and sufficient smoothness of the solution will always lead to a high consistency order.…”

“…Minimizing the quadratic form (19) over the weights a j (λ ) yields discretizations with optimal consistency with respect to the choice of the space U S [7]. But their calculation may be unstable [18] and they usually lead to non-sparse matrices unless users restrict the used nodes for each single functional.…”