476 research outputs found
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
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