16 research outputs found
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
Local RBF approximation for scattered data fitting with bivariate splines
In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation
In this paper we develop a discrete Hierarchical Basis (HB) to efficiently
solve the Radial Basis Function (RBF) interpolation problem with variable
polynomial order. The HB forms an orthogonal set and is adapted to the kernel
seed function and the placement of the interpolation nodes. Moreover, this
basis is orthogonal to a set of polynomials up to a given order defined on the
interpolating nodes. We are thus able to decouple the RBF interpolation problem
for any order of the polynomial interpolation and solve it in two steps: (1)
The polynomial orthogonal RBF interpolation problem is efficiently solved in
the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR
preconditioner. (2) The residual is then projected onto an orthonormal
polynomial basis. We apply our approach on several test cases to study its
effectiveness, including an application to the Best Linear Unbiased Estimator
regression problem
Error and Stability Estimates of the Least-Squares Variational Kernel-Based Methods for Second Order PDEs
We consider the least-squares variational kernel-based methods for numerical
solution of partial differential equations. Indeed, we focus on least-squares
principles to develop meshfree methods to find the numerical solution of a
general second order ADN elliptic boundary value problem in domain under Dirichlet boundary conditions. Most notably, in
these principles it is not assumed that differential operator is self-adjoint
or positive definite as it would have to be in the Rayleigh-Ritz setting.
However, the new scheme leads to a symmetric and positive definite algebraic
system allowing us to circumvent the compatibility conditions arising in
standard and mixed-Galerkin methods. In particular, the resulting method does
not require certain subspaces satisfying any boundary condition.
The trial space for discretization is provided via standard kernels that
reproduce , , as their native spaces. Therefore, the
smoothness of the approximation functions can be arbitrary increased without
any additional task. The solvability of the scheme is proved and the error
estimates are derived for functions in appropriate Sobolev spaces. For the
weighted discrete least-squares principles, we show that the optimal rate of
convergence in is accessible. Furthermore, for , the
proposed method has optimal rate of convergence in whenever . The condition number of the final linear system is approximated in
terms of discterization quality. Finally, the results of some computational
experiments support the theoretical error bounds.Comment: This paper includes 29 pages, 1 figure and 2 table