An adaptive multi-element generalized polynomial chaos method for stochastic differential equations

By Xiaoliang Wan, G. Em Karniadakis. 2005

636 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9617612891f3a04901f07965aeeb694b6edfd166

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In Civil and Structural Engineering

In Environmental Engineering

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Citations in the corpus

A least-squares approximation of partial differential equations with high-dimensional random inputs (2009, background): “Stochastic Galerkin schemes, [11,6,28,17,2,26], have been successfully applied to problems arising from different areas of engineering and are extremely useful for situations in which the number of uncertain parameters is not large.”
Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system (2013, methodology): “Working with a partitioned stochastic space The idea of using piecewise polynomial approximation of uncertain quantities has already been developed in the field of fluid mechanics [41, 42] for the study of limit cycles or equilibria.”
Asynchronous Time Integration for Polynomial Chaos Expansion of Uncertain Periodic Dynamics (2010, background): “In an effort to address this problem, [14] considered a multi-element generalized Polynomial Chaos approach.”
Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification (2013, background): “Krainchnan–Orszag three-mode problem Consider the system of ordinary differential equations [33]:”
Influence of Polynomial Chaos expansion order on an uncertain asymmetric rotor system response (2015, methodology): “Some recent works proposed to extend the Polynomial Chaos expansion by introducing the generalized Polynomial Chaos [6–8] or adaptive strategies based on the Multi-Element generalized Polynomial Chaos (MEgPC) [9].”
Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos (2013, methodology): “One possible way to account for this in the PCE is to apply a multielement approach [33], which will be addressed in future work.”
Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations (2006, methodology): “The only exception is the Legendre-chaos for the uniform distribution [19].”
Polynomial-Chaos-based Kriging (2015, positive): “This pio-neering approach called spectral stochastic ﬁnite element method (SSFEM), was later developed by Xiu and Karniadakis (2002); Sudret et al. (2004); Wan and Karniadakis (2005, 2006); Berveiller et al. (2006a), among others.”
A polynomial chaos expansion in dependent random variables (2018, methodology): “Approximations derived from truncated PCE, whether classical or generalized, are commonly used for solving uncertainty quantification problems, mostly in the context of solving stochastic partial differential equations [11,21], yielding approximate second-moment statistics of an output random variab”
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations (2009, methodology): “The multi-element generalized polynomial chaos method (MEgPC ) was introduced to address discontinuities in the random space while preserving the convergence rate of the gPC method [18] [19] [20].”
An ME-PC Enhanced HDMR Method for Efficient Statistical Analysis of Multiconductor Transmission Line Networks (2015, background): “become inefficient and inaccurate when the observable exhibits rapid variations and/or discontinuities [29], [30].”