34 research outputs found
Error estimates for interpolation of rough data using the scattered shifts of a radial basis function
The error between appropriately smooth functions and their radial basis
function interpolants, as the interpolation points fill out a bounded domain in
R^d, is a well studied artifact. In all of these cases, the analysis takes
place in a natural function space dictated by the choice of radial basis
function -- the native space. The native space contains functions possessing a
certain amount of smoothness. This paper establishes error estimates when the
function being interpolated is conspicuously rough.Comment: 12 page
Adaptive Density Estimation on the Circle by Nearly-Tight Frames
This work is concerned with the study of asymptotic properties of
nonparametric density estimates in the framework of circular data. The
estimation procedure here applied is based on wavelet thresholding methods: the
wavelets used are the so-called Mexican needlets, which describe a nearly-tight
frame on the circle. We study the asymptotic behaviour of the -risk
function for these estimates, in particular its adaptivity, proving that its
rate of convergence is nearly optimal.Comment: 30 pages, 3 figure
Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds
Let be a random sample from some unknown probability density
defined on a compact homogeneous manifold of dimension . Consider a 'needlet frame' describing a localised
projection onto the space of eigenfunctions of the Laplace operator on with corresponding eigenvalues less than , as constructed in
\cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform
deviations of the linear needlet density estimator obtained from an
empirical estimate of the needlet projection of . We apply these results to construct risk-adaptive
estimators and nonasymptotic confidence bands for the unknown density . The
confidence bands are adaptive over classes of differentiable and
H\"{older}-continuous functions on that attain their H\"{o}lder
exponents.Comment: Probability Theory and Related Fields, to appea
Imprints of the Quantum World in Classical Mechanics
The imprints left by quantum mechanics in classical (Hamiltonian) mechanics
are much more numerous than is usually believed. We show Using no physical
hypotheses) that the Schroedinger equation for a nonrelativistic system of
spinless particles is a classical equation which is equivalent to Hamilton's
equations.Comment: Paper submitted to Foundations of Physic
Quasi-Monte Carlo rules for numerical integration over the unit sphere
We study numerical integration on the unit sphere using equal weight quadrature rules, where the weights are such
that constant functions are integrated exactly.
The quadrature points are constructed by lifting a -net given in the
unit square to the sphere by means of an area
preserving map. A similar approach has previously been suggested by Cui and
Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2].
We prove three results. The first one is that the construction is (almost)
optimal with respect to discrepancies based on spherical rectangles. Further we
prove that the point set is asymptotically uniformly distributed on
. And finally, we prove an upper bound on the spherical cap
-discrepancy of order (where denotes the
number of points). This slightly improves upon the bound on the spherical cap
-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm.
Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the
-nets lifted to the sphere have spherical cap
-discrepancy converging with the optimal order of
Point sets on the sphere with small spherical cap discrepancy
In this paper we study the geometric discrepancy of explicit constructions of
uniformly distributed points on the two-dimensional unit sphere. We show that
the spherical cap discrepancy of random point sets, of spherical digital nets
and of spherical Fibonacci lattices converges with order . Such point
sets are therefore useful for numerical integration and other computational
simulations. The proof uses an area-preserving Lambert map. A detailed analysis
of the level curves and sets of the pre-images of spherical caps under this map
is given
On the stability of meshless symmetric collocation for boundary value problems
In this paper, we study the stability of symmetric collocation methods for boundary value problems using certain positive definite kernels. We derive lower bounds on the smallest eigenvalue of the associated collocation matrix in terms of the separation distance. Comparing these bounds to the well-known error estimates shows that another trade-off appears, which is significantly worse than the one known from classical interpolation. Finally, we show how this new trade-off can be overcome as well as how the collocation matrix can be stabilized by smoothing
Collocation discretizations of the transport equation with radial basis functions
In this paper, a new gridless method for numerically solving hyperbolic partial differential equations is presented. This method uses collocation based on Hermite interpolation at scattered sites at each time step. The basis can be chosen at each time step, which makes the approach adaptive and allows flexibility. The goal of this paper is to demonstrate the potential of this adaptive, flexible model which includes the ability to use scattered sites in a multi-dimensional setting with compactly supported radial functions. To illustrate the method and to compare it with known results, we analyze a certain transport problem in detail. When combined with an implicit time discretization and applied to smooth solutions, we show that the method yields spectral rates of convergence in the spatial domain. The implicit time discretization makes possible time steps of a size appropriate to the smoothness of the solution