Reconstruction of a discontinuous function from a few Fourier coefficients using Bayesian estimation
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.
scientific article; zbMATH DE number 849002
scientific article