Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
The nature of statistical learning theory
The nature of statistical learning theory
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Model Selection for Small Sample Regression
Machine Learning
A stochastic projection method for fluid flow II.: random process
Journal of Computational Physics
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Adaptive Generalized Polynomial Chaos for Nonlinear Random Oscillators
SIAM Journal on Scientific Computing
Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Beyond Wiener---Askey Expansions: Handling Arbitrary PDFs
Journal of Scientific Computing
Design and Analysis of Experiments
Design and Analysis of Experiments
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Generalized spectral decomposition for stochastic nonlinear problems
Journal of Computational Physics
Uncertainty quantification for algebraic systems of equations
Computers and Structures
Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis
Journal of Computational Physics
| Hi-index | 31.46 |
Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e. of Galerkin type) or non intrusive) unaffordable when the deterministic finite element model is expensive to evaluate. To address such problems, the paper describes a non intrusive method that builds a sparse PC expansion. First, an original strategy for truncating the PC expansions, based on hyperbolic index sets, is proposed. Then an adaptive algorithm based on least angle regression (LAR) is devised for automatically detecting the significant coefficients of the PC expansion. Beside the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon. The accuracy of the PC metamodel is checked using an estimate inspired by statistical learning theory, namely the corrected leave-one-out error. As a consequence, a rather small number of PC terms are eventually retained (sparse representation), which may be obtained at a reduced computational cost compared to the classical ''full'' PC approximation. The convergence of the algorithm is shown on an analytical function. Then the method is illustrated on three stochastic finite element problems. The first model features 10 input random variables, whereas the two others involve an input random field, which is discretized into 38 and 30-500 random variables, respectively.