Reconstruction of a discontinuous function from a few Fourier coefficients using Bayesian estimation
type: Item
instance of: scholarly article
title: Reconstruction of a discontinuous function from a few Fourier coefficients using Bayesian estimation
zbMATH Open document ID: 0840.65104
DOI: 10.1007/BF02087960
author: Alex Solomonoff
published in: Journal of Scientific Computing
publication date: 1996-06-30T00:00:00Z
review text: Spectral methods are applied to the numerical solution of conservation laws. It is assumed that the functions, which are solutions, have some probability distribution. Functions which are smooth away from discontinuities are assumed to be likely, and those which are not smooth away from discontinuities are assumed to be unlikely. Then a reconstruction algorithm is chosen by minimizing the expected error over all algorithms. It is possible to put the smoothness assumptions into an infinite-dimensional Gaussian probability distribution, and then the minimum-error algorithm is well known and simple to apply. If the Fourier coefficients of the reconstructed function are known exactly, then this approach gives very good results. But with the Fourier coefficients obtained from the spectral approximation of the Burgers equation the results are less impressive, probably because the coefficients are not known very accurately. It is shown that the performance of filters using Legendre and Chebyshev coefficients is similar to the Fourier case.
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35C10
Mathematics Subject Classification ID: 62A01
zbMATH DE Number: 849002
zbMATH Keywords: Fourier series solutions
zbMATH Keywords: Gibbs phenomenon
zbMATH Keywords: Bayesian analysis
zbMATH Keywords: spectral methods
zbMATH Keywords: conservation laws
zbMATH Keywords: reconstruction algorithm
zbMATH Keywords: Gaussian probability distribution
zbMATH Keywords: minimum-error algorithm
zbMATH Keywords: Burgers equation
reviewed by: Plamen Yalamov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 849002