Investigations on the restrictions of stochastic collocation methods for high dimensional and nonlinear engineering applications

authored by: Mona M. Dannert, Fynn Bensel, Amelie Fau, Rodolfo M.N. Fleury, Udo Nackenhorst
Abstract: Sophisticated sampling techniques used for solving stochastic partial differential equations efficiently and robustly are still in a state of development. It is known in the scientific community that global stochastic collocation methods using isotropic sparse grids are very efficient for simple problems but can become computationally expensive or even unstable for non-linear cases. The aim of this paper is to test the limits of these methods outside of a basic framework to provide a better understanding of their possible application in terms of engineering practices. Specifically, the stochastic collocation method using the Smolyak algorithm is applied to finite element problems with advanced features, such as high stochastic dimensions and non-linear material behaviour. We compare the efficiency and accuracy of different unbounded sparse grids (Gauss–Hermite, Gauss–Leja and Kronrod–Patterson) with Monte Carlo simulations. The sparse grids are constructed using an open source toolbox provided by Tamellini et al.,
Organisation(s): Institute of Mechanics and Computational Mechanics
International RTG 2657: Computational Mechanics Techniques in High Dimensions
External Organisation(s): École normale supérieure Paris-Saclay (ENS Paris-Saclay)
Altran Deutschland S.A.S.
Type: Article
Journal: Probabilistic Engineering Mechanics
Volume: 69
ISSN: 0266-8920
Publication date: 07.2022
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Civil and Structural Engineering, Nuclear Energy and Engineering, Condensed Matter Physics, Aerospace Engineering, Ocean Engineering, Mechanical Engineering
Electronic version(s): https://doi.org/10.1016/j.probengmech.2022.103299 (Access: Closed)

BibTeX

@article{4c5df1e587124823b8a566d1c261cd70,
title = "Investigations on the restrictions of stochastic collocation methods for high dimensional and nonlinear engineering applications",
abstract = "Sophisticated sampling techniques used for solving stochastic partial differential equations efficiently and robustly are still in a state of development. It is known in the scientific community that global stochastic collocation methods using isotropic sparse grids are very efficient for simple problems but can become computationally expensive or even unstable for non-linear cases. The aim of this paper is to test the limits of these methods outside of a basic framework to provide a better understanding of their possible application in terms of engineering practices. Specifically, the stochastic collocation method using the Smolyak algorithm is applied to finite element problems with advanced features, such as high stochastic dimensions and non-linear material behaviour. We compare the efficiency and accuracy of different unbounded sparse grids (Gauss–Hermite, Gauss–Leja and Kronrod–Patterson) with Monte Carlo simulations. The sparse grids are constructed using an open source toolbox provided by Tamellini et al.,",
keywords = "Karhunen–Lo{\`e}ve expansion, Modified exponential autocorrelation function, Non-linear stochastic finite element method, Plasticity, Random fields, Sparse grids, Stochastic collocation method",
author = "Dannert, {Mona M.} and Fynn Bensel and Amelie Fau and Fleury, {Rodolfo M.N.} and Udo Nackenhorst",
note = "Funding Information: This work has been partially funded by the German Research Foundation (DFG) during the priority program SPP 1886 ( NA330/12-1 ) which is gratefully acknowledged. The support of the French–German University is acknowledged under the French–German doctoral college “Sophisticated Numerical and Testing Approaches” (SNTA), grant DFDK 04-19 . This work was supported by the compute cluster, which is funded by the Leibniz University of Hannover, Germany , the Lower Saxony Ministry of Science and Culture (MWK), Germany and the German Research Foundation (DFG) . ",
year = "2022",
month = jul,
doi = "10.1016/j.probengmech.2022.103299",
language = "English",
volume = "69",
journal = "Probabilistic Engineering Mechanics",
issn = "0266-8920",
publisher = "Elsevier Ltd.",
}