Journal of Scientific Computing

, Volume 50, Issue 2, pp 306–340

Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection

  • Alen Alexanderian
  • Oliver P. Le Maître
  • Habib N. Najm
  • Mohamed Iskandarani
  • Omar M. Knio
Article

Abstract

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.

Keywords

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

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Alen Alexanderian
    • 1
  • Oliver P. Le Maître
    • 2
  • Habib N. Najm
    • 3
  • Mohamed Iskandarani
    • 4
  • Omar M. Knio
    • 1
  1. 1.Department of Mechanical EngineeringJohns Hopkins UniversityBaltimoreUSA
  2. 2.LIMSI-CNRSOrsay CedexFrance
  3. 3.Sandia National LaboratoriesLivermoreUSA
  4. 4.Rosenstiel School of Marine and Atmospheric ScienceUniversity of MiamiMiamiUSA

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