Abstract
An essential aspect of spectral representations analyzed so far is the projection of the process or solution on a polynomial basis, namely on a vector space spanned by infinitely differentiable functions. It is known following the works of Wiener [Am. J. Math. 60:897–936, 1938] and Cameron and Martin [Ann. Math. 48:385–392, 1947] that such spectral representation converges in a mean square sense as N,No→∞. In the case of numerical simulation, however, on must necessarily rely on truncation, and in this case the convergence may be seriously compromised due to truncation errors and aliasing, and in extreme cases these phenomena may lead to breakdown of the computations. This chapter explores the possibility of overcoming these difficulties by using a PC expansion based on Haar wavelets or multiwavelets.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Cameron, R., Martin, W.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)
Wiener, S.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Le Maître, O.P., Knio, O.M. (2010). Wavelet and Multiresolution Analysis Schemes. In: Spectral Methods for Uncertainty Quantification. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3520-2_8
Download citation
DOI: https://doi.org/10.1007/978-90-481-3520-2_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3519-6
Online ISBN: 978-90-481-3520-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)