Generalized spectral decomposition for stochastic nonlinear problems

• Published in: Journal of computational physics Vol. 228; no. 1; pp. 202 - 235
• Main Authors: Nouy, Anthony, Le Maıˆtre, Olivier P.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 10-01-2009; Elsevier
• Subjects: ALGORITHMS; APPROXIMATIONS; Burgers equation; CHAOS THEORY; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational techniques; CONVERGENCE; DECOMPOSITION; Eigenproblem; EQUATIONS; Exact sciences and technology; Galerkin methods; Generalized spectral decomposition; Mathematical analysis; Mathematical methods in physics; Mathematical models; MATHEMATICAL SOLUTIONS; Nonlinear problem; NONLINEAR PROBLEMS; Nonlinearity; ONE-DIMENSIONAL CALCULATIONS; Physics; POLYNOMIALS; RESOLUTION; Spectra; STOCHASTIC PROCESSES; Stochastic spectral decompositions; Stochasticity; TWO-DIMENSIONAL CALCULATIONS; Uncertainty quantification; Uncertainty quantification; Generalized spectral decomposition; Eigenproblem; Stochastic spectral decompositions; Nonlinear problem; Chaos; Uncertainty; Spectral method; Decomposition method; Stochastic approximation; Nonlinear problems; Burgers equation; Calculation methods; Galerkin-Petrov method; Wavelets; Algorithms; Discretization; Convergence rate; Models; Calculation; Generalized special decomposition; Diffusion
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2008.09.010

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.