17,066 research outputs found
Approximate Approximations from scattered data
The aim of this paper is to extend the approximate quasi-interpolation on a
uniform grid by dilated shifts of a smooth and rapidly decaying function on a
uniform grid to scattered data quasi-interpolation. It is shown that high order
approximation of smooth functions up to some prescribed accuracy is possible,
if the basis functions, which are centered at the scattered nodes, are
multiplied by suitable polynomials such that their sum is an approximate
partition of unity. For Gaussian functions we propose a method to construct the
approximate partition of unity and describe the application of the new
quasi-interpolation approach to the cubature of multi-dimensional integral
operators.Comment: 29 pages, 17 figure
Approximate approximations from scattered data
AbstractThe aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators
On the convergence of the rescaled localized radial basis function method
The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture
The Penalized Lebesgue Constant for Surface Spline Interpolation
Problems involving approximation from scattered data where data is arranged
quasi-uniformly have been treated by RBF methods for decades. Treating data
with spatially varying density has not been investigated with the same
intensity, and is far less well understood. In this article we consider the
stability of surface spline interpolation (a popular type of RBF interpolation)
for data with nonuniform arrangements. Using techniques similar to those
recently employed by Hangelbroek, Narcowich and Ward to demonstrate the
stability of interpolation from quasi-uniform data on manifolds, we show that
surface spline interpolation on R^d is stable, but in a stronger, local sense.
We also obtain pointwise estimates showing that the Lagrange function decays
very rapidly, and at a rate determined by the local spacing of datasites. These
results, in conjunction with a Lebesgue lemma, show that surface spline
interpolation enjoys the same rates of convergence as those of the local
approximation schemes recently developed by DeVore and Ron.Comment: 20 pages; corrected typos; to appear in Proc. Amer. Math. So
Local interpolation schemes for landmark-based image registration: a comparison
In this paper we focus, from a mathematical point of view, on properties and
performances of some local interpolation schemes for landmark-based image
registration. Precisely, we consider modified Shepard's interpolants,
Wendland's functions, and Lobachevsky splines. They are quite unlike each
other, but all of them are compactly supported and enjoy interesting
theoretical and computational properties. In particular, we point out some
unusual forms of the considered functions. Finally, detailed numerical
comparisons are given, considering also Gaussians and thin plate splines, which
are really globally supported but widely used in applications
- …