Generalized spectral decomposition for stochastic nonlinear problems

Anthony Nouy, Olivier P. Le Maître*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

94 Scopus citations

Abstract

We present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multi-wavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low-dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional nonlinear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space.

Original languageEnglish (US)
Pages (from-to)202-235
Number of pages34
JournalJournal of Computational Physics
Volume228
Issue number1
DOIs
StatePublished - Jan 10 2009

Keywords

  • Eigenproblem
  • Generalized spectral decomposition
  • Nonlinear problem
  • Stochastic spectral decompositions
  • 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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