Range restricted C2 interpolant to scattered data

The construction of a range restricted bivariate C2 interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. Sufficient conditions are derived on Bezier points in order to ensure that surfaces comprising quintic Bezier triangular patches are always positive and satisfy C2 continuity conditions. The first and second derivatives at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Its construction is local and easily extended to include as upper and lower constraints to the interpolating surfaces of the form z = C(x,y) where C is a polynomial of degree less or equal to 5. A number of examples are presented

Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page

RBF approximation of large datasets by partition of unity and local stabilization

We present an algorithm to approximate large dataset by Radial Basis Function (RBF) techniques. The method couples a fast domain decomposition procedure with a localized stabilization method. The resulting algorithm can efficiently deal with large problems and it is robust with respect to the typical instability of kernel methods

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

Range Restricted Interpolation Using Cubic Bézier Triangles.

A range restricted C1 interpolation local scheme to scattered data is derived. Each macro triangle of the triangulated domain is split into three mini triangles and the interpolating surface on each mini triangle is a cubic Bézier triangle. Sufficient conditions derived for the non-negativity of these cubic Bézier triangles are expressed as lower bounds to the Bézier ordinates. The non-negativity preserving interpolation scheme extends to the construction of a range restricted interpolating surface with lower or upper constraints which are polynomial surfaces of degree up to three. The scheme is illustrated with graphical examples