Multiscale stochastic preconditioners in non-intrusive spectral projection

Alen Alexanderian, Oliver P. Le Maître, Habib N. Najm, Mohamed Iskandarani, Omar M. Knio*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Scopus citations


A preconditioning approach is developed that enables efficient polynomial chaos (PC) representations of uncertain dynamical systems. The approach is based on the definition of an appropriate multiscale stretching of the individual components of the dynamical system which, in particular, enables robust recovery of the unscaled transient dynamics. Efficient PC representations of the stochastic dynamics are then obtained through non-intrusive spectral projections of the stretched measures. Implementation of the present approach is illustrated through application to a chemical system with large uncertainties in the reaction rate constants. Computational experiments show that, despite the large stochastic variability of the stochastic solution, the resulting dynamics can be efficiently represented using sparse low-order PC expansions of the stochastic multiscale preconditioner and of stretched variables. The present experiences are finally used to motivate several strategies that promise to yield further advantages in spectral representations of stochastic dynamics.

Original languageEnglish (US)
Pages (from-to)306-340
Number of pages35
JournalJournal of Scientific Computing
Issue number2
StatePublished - Feb 2012
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgements Research supported by the US Department of Energy, Office of Advanced Scientific Computing Research under Award DE-SC0001980 (AA, OMK), by Office of Naval Research under Award N00014-10-1-0498 (AA, OMK, MI), and by the French National Research Agency Grant ANR-08-JCJC-0022 (OLM). HNN acknowledges the support of the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences. Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US DOE under contract DE-AC04-94-AL85000.


  • Non-intrusive spectral projection
  • Polynomial chaos
  • Stochastic preconditioner
  • Stretched measure
  • Uncertain dynamical system

ASJC Scopus subject areas

  • Software
  • Engineering(all)
  • Computational Mathematics
  • Theoretical Computer Science
  • Applied Mathematics
  • Numerical Analysis
  • Computational Theory and Mathematics


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