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


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

ASJC Scopus subject areas

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


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