Efficient computation with radial basis functions

The proposal is concerned with developing more effective computational techniques for the reconstruction of information when the data has little structure and may be in high space dimension. Radial basis function (RBF) methods have proved a popular choice for this task. We intend to continue the development of cheap preconditioning methods for RBFs and provide theory and numerical experiments to support efficient computational methods for reconstruction when data is anisotropic.

New error bounds have been developed for the interpolation with anisotropically transformed radial basis functions that predict a significant improvement of the approximation error if both the data behaviour and the placement of the interpolation centres are anisotropic. This improvement is confirmed numerically. Though practical work has been done using such a strategy, and work on the conditioning of the interpolation matrices has been completed, these are the first known results giving error bounds for such an approximation scheme. In particular, we produced results which mirror similar results from the finite element literature. The anisotropic linear transformation is best used in conjunction with a two-stage approximation process, where the linear transformations are used locally to obtain high quality approximations on subdomains, and the global approximation is produced using either a partition of unity, or a piecewise polynomial quasi-interpolant. A significant progress has been achieved in the development of a novel two-stage method of the second type, where the spline space is constructed with respect to a multiresolution sequence of adaptively refined triangulations and the local approximations are computed with the help of anisotropically transformed radial basis functions, with adaptively chosen local transformations. We expect that this approach will produce improved for the problem of scattered data fitting which is important for many practical applications.