Approximation on the sphere using radial basis functions plus polynomials

Abstract

In this paper we analyse a hybrid approximation of functions on the sphere S^2 ⊂ R^3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation

Inf-sup condition for spherical polynomials and radial basis functions on spheres

“…In this section we illustrate the use of the inf-sup condition by using it to give a new error analysis for the hybrid polynomial-plus-radial-basis-function method of Sloan and Sommariva [11]. We shall assume in this section that Φ = Φ s is the reproducing kernel for the space H s given by (2).…”

“…In Section 5 we use the inf-sup condition to obtain a new error analysis, this time for the L 2 norm of the error, for the hybrid interpolation scheme of Sloan and Sommariva [11]. Finally, in Section 6 the results of the paper are extended to approximation with more general kernels Φ(x, y) for which the norm in the associated "native space" is merely equivalent (rather then identical) to the H s norm.…”

“…In this paper we prove an inf-sup condition in a completely different context, that of approximation by spherical polynomials and radial basis functions on spheres of arbitrary dimension, as discussed in [11].…”

Boundary integral equations on the sphere with radial basis functions: error analysis

“…This is a result of Proposition 3.2 (taking w = I X v − v), of (3.9) and the equivalence of the native space norm and H τ -norm. 2 When v is smoother, the error bound can be improved using the technique developed in [11] and [15], which is modified in [13] for hybrid approximation using radial basis functions and polynomials. The crucial tool is the isomorphism defined in the following lemma.…”

Natural Preconditioning and Iterative Methods for Saddle Point Systems

“…by hybrid methods that combine local radial basis functions with global spherical polynomials [63,70]. To ensure that the interpolant is unique, the radial basis function component is constrained to be orthogonal to the spherical polynomial approximation space with respect to the inner product associated with the radial basis functions (the native space inner product).…”