2,660 research outputs found
Monotonicity preserving approximation of multivariate scattered data
This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /
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
Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples
We present a new approach to three-dimensional electromagnetic scattering
problems via fast isogeometric boundary element methods. Starting with an
investigation of the theoretical setting around the electric field integral
equation within the isogeometric framework, we show existence, uniqueness, and
quasi-optimality of the isogeometric approach. For a fast and efficient
computation, we then introduce and analyze an interpolation-based fast
multipole method tailored to the isogeometric setting, which admits competitive
algorithmic and complexity properties. This is followed by a series of
numerical examples of industrial scope, together with a detailed presentation
and interpretation of the results
Surface Spline Approximation on SO(3)
The purpose of this article is to introduce a new class of kernels on SO(3)
for approximation and interpolation, and to estimate the approximation power of
the associated spaces. The kernels we consider arise as linear combinations of
Green's functions of certain differential operators on the rotation group. They
are conditionally positive definite and have a simple closed-form expression,
lending themselves to direct implementation via, e.g., interpolation, or
least-squares approximation. To gauge the approximation power of the underlying
spaces, we introduce an approximation scheme providing precise L_p error
estimates for linear schemes, namely with L_p approximation order conforming to
the L_p smoothness of the target function.Comment: 22 pages, to appear in Appl. Comput. Harmon. Ana
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