Although most work to date on RBFs relates to scattered data approximation and in general to interpolation theory, there has recently been an increased interest in their use for solving partial differential equations (PDEs). This approach, which approximates the whole solution of the PDE directly using RBFs, is very attractive due to the fact that this is truly a mesh-free technique. Kansa [1] ...

In this paper we consider the problem of developing a variational theory for interpolation by radial basis functions on spheres. The interpolants have the property that they minimise the value of a certain semi-norm, which we construct explicitly. We then go on to investigate forms of the interpolant which are suitable for computation. Our main aim is to derive error bounds for interpolation fr...

In this paper we study the method of interpolation by radial basis functions and give some error estimates in Sobolev space H(Ω) (k ≥ 1). With a special kind of radial basis function, we construct a basis in H(Ω) and derive a meshless method for solving elliptic partial differential equations. We also propose a method for computing the global data density. Mathematics subject classification: 41...

While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline ...

Interpolation using radial basis functions has become an established and useful technique in multi-dimensional data tting. It is a special case of interpolation by translates of a basis function. All the methods used in practice have a strong connection with a variational principle. In this paper we provide a straightforward approach to the variational principle, and make some useful practical ...

A multilevel sparse kernel-based interpolation (MLSKI) method, suitable for moderately high-dimensional function interpolation problems has been recently proposed in [15]. The method uses both level-wise and direction-wise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling...

We give an explicit example for the selection of the shape parameter for a certain univariate radial basis function (RBF) interpolation problem.