Multi-resolution analysis of Wiener-type uncertainty propagation schemes

O. P. Le Maître, H. N. Najm, R. G. Ghanem, O. M. Knio*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

281 Scopus citations


A multi-resolution analysis (MRA) is applied to an uncertainty propagation scheme based on a generalized polynomial chaos (PC) representation. The MRA relies on an orthogonal projection of uncertain data and solution variables onto a multi-wavelet basis, consisting of compact piecewise-smooth polynomial functions. The coefficients of the expansion are computed through a Galerkin procedure. The MRA scheme is applied to the simulation of the Lorenz system having a single random parameter. The convergence of the solution with respect to the resolution level and expansion order is investigated. In particular, results are compared to two Monte-Carlo sampling strategies, demonstrating the superiority of the MRA. For more complex problems, however, the MRA approach may require excessive CPU times. Adaptive methods are consequently developed in order to overcome this drawback. Two approaches are explored: the first is based on adaptive refinement of the multi-wavelet basis, while the second is based on adaptive block-partitioning of the space of random variables. Computational tests indicate that the latter approach is better suited for large problems, leading to a more efficient, flexible and parallelizable scheme.

Original languageEnglish (US)
Pages (from-to)502-531
Number of pages30
JournalJournal of Computational Physics
Issue number2
StatePublished - Jul 1 2004
Externally publishedYes

Bibliographical note

Funding Information:
This work was supported by the Laboratory Directed Research and Development Program at Sandia National Laboratories, funded by the US Department of Energy and by the DOE office of Basic Energy Sciences, Chemical Sciences Division. Support was also provided by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-00-2-0612. The US government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Computations were performed at the National Center for Supercomputer Applications.


  • Adaptive scheme
  • Multi-resolution analysis
  • Multi-wavelets
  • Polynomial chaos
  • Uncertainty quantification

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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