15,404 research outputs found
ENO-wavelet transforms for piecewise smooth functions
We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate
Wavelet treatment of the intra-chain correlation functions of homopolymers in dilute solutions
Discrete wavelets are applied to parametrization of the intra-chain two-point
correlation functions of homopolymers in dilute solutions obtained from Monte
Carlo simulation. Several orthogonal and biorthogonal basis sets have been
investigated for use in the truncated wavelet approximation. Quality of the
approximation has been assessed by calculation of the scaling exponents
obtained from des Cloizeaux ansatz for the correlation functions of
homopolymers with different connectivities in a good solvent. The resulting
exponents are in a better agreement with those from the recent renormalisation
group calculations as compared to the data without the wavelet denoising. We
also discuss how the wavelet treatment improves the quality of data for
correlation functions from simulations of homopolymers at varied solvent
conditions and of heteropolymers.Comment: RevTeX, 19 pages, 7 PS figures. Accepted for publication in PR
A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets
In nonparametric statistical problems, we wish to find an estimator of an
unknown function f. We can split its error into bias and variance terms;
Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel
estimate, the supremum norm of the variance term is asymptotically distributed
as a Gumbel random variable. In the following, we prove a version of this
result for estimators using compactly-supported wavelets, a popular tool in
nonparametric statistics. Our result relies on an assumption on the nature of
the wavelet, which must be verified by provably-good numerical approximations.
We verify our assumption for Daubechies wavelets and symlets, with N = 6, ...,
20 vanishing moments; larger values of N, and other wavelet bases, are easily
checked, and we conjecture that our assumption holds also in those cases
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