5,530 research outputs found
Subperiodic trigonometric subsampling: A numerical approach
We show that Gauss-Legendre quadrature applied to trigonometric poly- nomials on subintervals of the period can be competitive with subperiodic trigonometric Gaussian quadrature. For example with intervals correspond- ing to few angular degrees, relevant for regional scale models on the earth surface, we see a subsampling ratio of one order of magnitude already at moderate trigonometric degrees
A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces
We present a simple, accurate method for computing singular or nearly
singular integrals on a smooth, closed surface, such as layer potentials for
harmonic functions evaluated at points on or near the surface. The integral is
computed with a regularized kernel and corrections are added for regularization
and discretization, which are found from analysis near the singular point. The
surface integrals are computed from a new quadrature rule using surface points
which project onto grid points in coordinate planes. The method does not
require coordinate charts on the surface or special treatment of the
singularity other than the corrections. The accuracy is about , where
is the spacing in the background grid, uniformly with respect to the point
of evaluation, on or near the surface. Improved accuracy is obtained for points
on the surface. The treecode of Duan and Krasny for Ewald summation is used to
perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy
Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors
We determine optimal designs for some regression models which are frequently
used for describing three-dimensional shapes. These models are based on a
Fourier expansion of a function defined on the unit sphere in terms of
spherical harmonic basis functions. In particular, it is demonstrated that the
uniform distribution on the sphere is optimal with respect to all
criteria proposed by Kiefer in 1974 and also optimal with respect to a
criterion which maximizes a mean of the smallest eigenvalues of the
variance--covariance matrix. This criterion is related to principal component
analysis, which is the common tool for analyzing this type of image data.
Moreover, discrete designs on the sphere are derived, which yield the same
information matrix in the spherical harmonic regression model as the uniform
distribution and are therefore directly implementable in practice. It is
demonstrated that the new designs are substantially more efficient than the
commonly used designs in three-dimensional shape analysis.Comment: Published at http://dx.doi.org/10.1214/009053605000000552 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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