Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Stochastic partial differential equations: a modeling, white noise functional approach
Stochastic partial differential equations: a modeling, white noise functional approach
Advances in Engineering Software - Special issue on large-scale analysis, design and intelligent synthesis environments
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
A stochastic projection method for fluid flow II.: random process
Journal of Computational Physics
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure
SIAM Journal on Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems
SIAM Journal on Scientific Computing
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
Hierarchical parallelisation for the solution of stochastic finite element equations
Computers and Structures
Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows
Journal of Computational Physics
Proper general decomposition (PGD) for the resolution of Navier-Stokes equations
Journal of Computational Physics
Adaptive sparse polynomial chaos expansion based on least angle regression
Journal of Computational Physics
Sampling-free linear Bayesian update of polynomial chaos representations
Journal of Computational Physics
A model reduction technique based on the PGD for elastic-viscoplastic computational analysis
Computational Mechanics
Journal of Computational Physics
Hi-index | 31.48 |
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.