Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach

Géraud Blatman; Bruno Sudret

doi:10.1016/j.crme.2008.02.013

Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach
[Chaos polynomial creux et éléments finis stochastiques adaptatifs : une approche par régression]

Présenté par : Jean-Baptiste Leblond

Géraud Blatman ^1,² ; Bruno Sudret ²

¹ IFMA-LaMI, Campus des Cézeaux, BP 265, 63175 Aubière cedex, France
² EDF R&D, Département matériaux et mécanique des composants, site des Renardières, 77250 Moret-sur-Loing cedex, France

Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 518-523.

Résumé
Abstract

Dans cette communication, on propose un algorithme permettant de construire une représentation par chaos polynomial creux de la réponse d'un modèle mécanique dont les paramètres d'entrée sont aléatoires. L'algorithme construit de façon adaptative une représentation creuse en détectant automatiquement les termes importants et en supprimant ceux qui sont négligeables. A chaque étape, le calcul des coefficients s'effectue par minimisation au sens des moindres carrés (méthode non-intrusive dite de régression). L'algorithme est déroulé pas à pas sur un modèle polynomial, puis appliqué à l'étude de la fiabilité d'un treillis élastique.

A method is proposed to build a sparse polynomial chaos (PC) expansion of a mechanical model whose input parameters are random. In this respect, an adaptive algorithm is described for automatically detecting the significant coefficients of the PC expansion. The latter can thus be computed by means of a relatively small number of possibly costly model evaluations, using a non-intrusive regression scheme (also known as stochastic collocation). The method is illustrated by a simple polynomial model, as well as the example of the deflection of a truss structure.

Reçu le : 2008-01-18
Accepté le : 2008-02-25
Publié le : 2008-04-28

DOI : 10.1016/j.crme.2008.02.013

Keywords: Solids and structures, Adaptive stochastic finite elements, Sparse polynomial chaos, Stochastic collocation, Regression, Structural reliability
Mots-clés : Solides et structures, Eléments finis stochastiques adaptatifs, Chaos polynomial creux, Collocation stochastique, Régression, Fiabilité

Affiliations des auteurs :

Géraud Blatman ^{1, 2} ; Bruno Sudret ²

¹ IFMA-LaMI, Campus des Cézeaux, BP 265, 63175 Aubière cedex, France
² EDF R&D, Département matériaux et mécanique des composants, site des Renardières, 77250 Moret-sur-Loing cedex, France

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     doi = {10.1016/j.crme.2008.02.013},
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Géraud Blatman; Bruno Sudret. Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus. Mécanique, Volume 336 (2008) no. 6, pp. 518-523. doi : 10.1016/j.crme.2008.02.013. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.02.013/

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Vinh Ngoc Tran; Jongho Kim Toward an Efficient Uncertainty Quantification of Streamflow Predictions Using Sparse Polynomial Chaos Expansion, Water, Volume 13 (2021) no. 2, p. 203 | DOI:10.3390/w13020203
Sayed Hamid Sohrabi; Mohammad Javad Ketabdari Stochastic modeling and sensitivity analysis of underwater sound absorber rubber coating, Applied Acoustics, Volume 164 (2020), p. 107282 | DOI:10.1016/j.apacoust.2020.107282
Xiangfeng Guo; Qiangqiang Sun; Daniel Dias; Eric Antoinet Probabilistic assessment of an earth dam stability design using the adaptive polynomial chaos expansion, Bulletin of Engineering Geology and the Environment, Volume 79 (2020) no. 9, p. 4639 | DOI:10.1007/s10064-020-01847-2
Dinesh Kumar; Yao Koutsawa; Gaston Rauchs; Mariapia Marchi; Carlos Kavka; Salim Belouettar Efficient uncertainty quantification and management in the early stage design of composite applications, Composite Structures, Volume 251 (2020), p. 112538 | DOI:10.1016/j.compstruct.2020.112538
Sergey Oladyshkin; Farid Mohammadi; Ilja Kroeker; Wolfgang Nowak Bayesian3 Active Learning for the Gaussian Process Emulator Using Information Theory, Entropy, Volume 22 (2020) no. 8, p. 890 | DOI:10.3390/e22080890
Edoardo Menga; María J. Sánchez; Ignacio Romero Anisotropic meta-models for computationally expensive simulations in nonlinear mechanics, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 5, pp. 904-924 | DOI:10.1002/nme.6250 | Zbl:1548.74008
T. D. Dao; Q. Serra; S. Berger; E. Florentin Error estimation of polynomial chaos approximations in transient structural dynamics, International Journal of Computational Methods, Volume 17 (2020) no. 10, p. 23 (Id/No 2050003) | DOI:10.1142/s0219876220500036 | Zbl:1550.74465
Han Wang; Zheng Yan; Xiaoyuan Xu; Kun He Probabilistic power flow analysis of microgrid with renewable energy, International Journal of Electrical Power Energy Systems, Volume 114 (2020), p. 105393 | DOI:10.1016/j.ijepes.2019.105393
Biswarup Bhattacharyya Global sensitivity analysis: a Bayesian learning based polynomial chaos approach, Journal of Computational Physics, Volume 415 (2020), p. 21 (Id/No 109539) | DOI:10.1016/j.jcp.2020.109539 | Zbl:1440.62093
Jakob Dürrwächter; Thomas Kuhn; Fabian Meyer; Louisa Schlachter; Florian Schneider A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations, Journal of Computational and Applied Mathematics, Volume 370 (2020), p. 22 (Id/No 112602) | DOI:10.1016/j.cam.2019.112602 | Zbl:1450.65118
N Afanador García; C J Noriega Sanchez; G Guerrero G Quantification of uncertainty in steel plates subject to fatigue with variable load via chaos expansion polynomial, Journal of Physics: Conference Series, Volume 1708 (2020) no. 1, p. 012032 | DOI:10.1088/1742-6596/1708/1/012032
Hayley Guy; Alen Alexanderian; Meilin Yu A distributed active subspace method for scalable surrogate modeling of function valued outputs, Journal of Scientific Computing, Volume 85 (2020) no. 2, p. 24 (Id/No 36) | DOI:10.1007/s10915-020-01346-2 | Zbl:1453.65009
Gaël Poëtte Spectral convergence of the generalized polynomial chaos reduced model obtained from the uncertain linear Boltzmann equation, Mathematics and Computers in Simulation, Volume 177 (2020), pp. 24-45 | DOI:10.1016/j.matcom.2020.04.009 | Zbl:1510.65273
Yicheng Zhou; Zhenzhou Lu; Wanying Yun Active sparse polynomial chaos expansion for system reliability analysis, Reliability Engineering System Safety, Volume 202 (2020), p. 107025 | DOI:10.1016/j.ress.2020.107025
Qiuqi Li; Pingwen Zhang A variable-separation method for nonlinear partial differential equations with random inputs, SIAM Journal on Scientific Computing, Volume 42 (2020) no. 2, p. a723-a750 | DOI:10.1137/19m1262486 | Zbl:1432.35266
Marc Mignolet; Christian Soize Compressed principal component analysis of non-Gaussian vectors, SIAM/ASA Journal on Uncertainty Quantification, Volume 8 (2020), pp. 1261-1286 | DOI:10.1137/20m1322029 | Zbl:1451.62069
M.-H. Trinh; S. Berger; E. Aubry Nonlinear Dynamic Behaviour Analysis of a Clutch System with Uncertainties Using Polynomial Chaos and the Constrained Harmonic Balance Method, Shock and Vibration, Volume 2020 (2020), p. 1 | DOI:10.1155/2020/8401745
Zeping Wu; Donghui Wang; Wenjie Wang; Kun Zhao; Houcun Zhou; Weihua Zhang Hybrid metamodel of radial basis function and polynomial chaos expansions with orthogonal constraints for global sensitivity analysis, Structural and Multidisciplinary Optimization, Volume 62 (2020) no. 2, p. 597 | DOI:10.1007/s00158-020-02516-4
Jiangjiang Zhang; Qiang Zheng; Dingjiang Chen; Laosheng Wu; Lingzao Zeng Surrogate‐Based Bayesian Inverse Modeling of the Hydrological System: An Adaptive Approach Considering Surrogate Approximation Error, Water Resources Research, Volume 56 (2020) no. 1 | DOI:10.1029/2019wr025721
Vinh Ngoc Tran; M. Chase Dwelle; Khachik Sargsyan; Valeriy Y. Ivanov; Jongho Kim A Novel Modeling Framework for Computationally Efficient and Accurate Real‐Time Ensemble Flood Forecasting With Uncertainty Quantification, Water Resources Research, Volume 56 (2020) no. 3 | DOI:10.1029/2019wr025727
Yicheng Zhou; Zhenzhou Lu Active Polynomial Chaos Expansion for Reliability-Based Design Optimization, AIAA Journal, Volume 57 (2019) no. 12, p. 5431 | DOI:10.2514/1.j058020
Justin Weinmeister; Xinfeng Gao; Sourajeet Roy Analysis of a Polynomial Chaos-Kriging Metamodel for Uncertainty Quantification in Aerodynamics, AIAA Journal, Volume 57 (2019) no. 6, p. 2280 | DOI:10.2514/1.j057527
M. Chase Dwelle; Jongho Kim; Khachik Sargsyan; Valeriy Y. Ivanov Streamflow, stomata, and soil pits: Sources of inference for complex models with fast, robust uncertainty quantification, Advances in Water Resources, Volume 125 (2019), p. 13 | DOI:10.1016/j.advwatres.2019.01.002
K. Dammak; S. Koubaa; A. El Hami; L. Walha; M. Haddar Numerical modelling of vibro-acoustic problem in presence of uncertainty: Application to a vehicle cabin, Applied Acoustics, Volume 144 (2019), p. 113 | DOI:10.1016/j.apacoust.2017.06.001
Xiao Wei; Haichao Chang; Baiwei Feng; Zuyuan Liu; Chenran Huang Hull form reliability-based robust design optimization combining polynomial chaos expansion and maximum entropy method, Applied Ocean Research, Volume 90 (2019), p. 101860 | DOI:10.1016/j.apor.2019.101860
Markus Köppel; Fabian Franzelin; Ilja Kröker; Sergey Oladyshkin; Gabriele Santin; Dominik Wittwar; Andrea Barth; Bernard Haasdonk; Wolfgang Nowak; Dirk Pflüger; Christian Rohde Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario, Computational Geosciences, Volume 23 (2019) no. 2, pp. 339-354 | DOI:10.1007/s10596-018-9785-x | Zbl:1414.76058
Yicheng Zhou; Zhenzhou Lu; Kai Cheng; Chunyan Ling An efficient and robust adaptive sampling method for polynomial chaos expansion in sparse Bayesian learning framework, Computer Methods in Applied Mechanics and Engineering, Volume 352 (2019), pp. 654-674 | DOI:10.1016/j.cma.2019.04.046 | Zbl:1441.62201
Xiangfeng Guo; Daniel Dias; Qiujing Pan Probabilistic stability analysis of an embankment dam considering soil spatial variability, Computers and Geotechnics, Volume 113 (2019), p. 103093 | DOI:10.1016/j.compgeo.2019.103093
Xiangfeng Guo; Daniel Dias; Claudio Carvajal; Laurent Peyras; Pierre Breul A comparative study of different reliability methods for high dimensional stochastic problems related to earth dam stability analyses, Engineering Structures, Volume 188 (2019), p. 591 | DOI:10.1016/j.engstruct.2019.03.056
Xiangfeng Guo; Dianchun Du; Daniel Dias Reliability analysis of tunnel lining considering soil spatial variability, Engineering Structures, Volume 196 (2019), p. 109332 | DOI:10.1016/j.engstruct.2019.109332
Rocío Rodríguez‐Cantano; Joakim Sundnes; Marie E. Rognes Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response, International Journal for Numerical Methods in Biomedical Engineering, Volume 35 (2019) no. 5 | DOI:10.1002/cnm.3178
Hongguan Zhang; Tadahiro Shibutani Development of stochastic isogeometric analysis (SIGA) method for uncertainty in shape, International Journal for Numerical Methods in Engineering, Volume 118 (2019) no. 1, pp. 18-37 | DOI:10.1002/nme.6008 | Zbl:1548.74960
Kamaljyoti Nath; Anjan Dutta; Budhaditya Hazra An iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non-Gaussian randomness, International Journal for Numerical Methods in Engineering, Volume 119 (2019) no. 11, pp. 1126-1160 | DOI:10.1002/nme.6086 | Zbl:1548.74968
Yicheng Zhou; Zhenzhou Lu; Kai Cheng A new surrogate modeling method combining polynomial chaos expansion and Gaussian kernel in a sparse Bayesian learning framework, International Journal for Numerical Methods in Engineering, Volume 120 (2019) no. 4, pp. 498-516 | DOI:10.1002/nme.6145 | Zbl:1548.60028
Nassim Kernou; Youcef Bouafia Development of New Approach in Reliability Analysis for Excellent Predictive Quality of the Approximation Using Adaptive Kriging, International Journal of Engineering Research in Africa, Volume 44 (2019), p. 44 | DOI:10.4028/www.scientific.net/jera.44.44
Gaël Poëtte A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation, Journal of Computational Physics, Volume 385 (2019), pp. 135-162 | DOI:10.1016/j.jcp.2019.01.052 | Zbl:1451.65006
Kamaljyoti Nath; Anjan Dutta; Budhaditya Hazra An iterative polynomial chaos approach for solution of structural mechanics problem with Gaussian material property, Journal of Computational Physics, Volume 390 (2019), pp. 425-451 | DOI:10.1016/j.jcp.2019.04.014 | Zbl:1452.74109
Jian Song; Yun Yang; Gan Chen; Xiaomin Sun; Jin Lin; Jianfeng Wu; Jichun Wu Surrogate assisted multi-objective robust optimization for groundwater monitoring network design, Journal of Hydrology, Volume 577 (2019), p. 123994 | DOI:10.1016/j.jhydrol.2019.123994
N Afanador García Quantification of uncertainty in metallic elements subjected to fatigue, Journal of Physics: Conference Series, Volume 1329 (2019) no. 1, p. 012011 | DOI:10.1088/1742-6596/1329/1/012011
Manav Vohra; Alen Alexanderian; Cosmin Safta; Sankaran Mahadevan Sensitivity-driven adaptive construction of reduced-space surrogates, Journal of Scientific Computing, Volume 79 (2019) no. 2, pp. 1335-1359 | DOI:10.1007/s10915-018-0894-4 | Zbl:1419.65006
Yicheng Zhou; Zhenzhou Lu; Kai Cheng; Yan Shi An expanded sparse Bayesian learning method for polynomial chaos expansion, Mechanical Systems and Signal Processing, Volume 128 (2019), p. 153 | DOI:10.1016/j.ymssp.2019.03.032
Duc Thinh Kieu; Baptiste Bergeot; Marie-Laure Gobert; Sébastien Berger Stability analysis of a clutch system with uncertain parameters using sparse polynomial chaos expansions, Mechanics Industry, Volume 20 (2019) no. 1, p. 104 | DOI:10.1051/meca/2019003
Fatma Abid; Tarek Merzouki; Abdelkhalak El Hami; Hassen Trabelsi; Lassaad Walha; Mohamed Haddar Uncertainty of shape memory alloy micro-actuator using generalized polynomial chaos method, Microsystem Technologies, Volume 25 (2019) no. 4, p. 1505 | DOI:10.1007/s00542-018-4199-1
I.V. González; M.A. Valdebenito; J.I. Correa; H.A. Jensen Calculation of second order statistics of uncertain linear systems applying reduced order models, Reliability Engineering System Safety, Volume 190 (2019), p. 106514 | DOI:10.1016/j.ress.2019.106514
Yicheng Zhou; Zhenzhou Lu; Kai Cheng Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation, Structural and Multidisciplinary Optimization, Volume 59 (2019) no. 1, p. 229 | DOI:10.1007/s00158-018-2062-8
Mingang Yin; Jian Wang; Zhili Sun An innovative DoE strategy of the kriging model for structural reliability analysis, Structural and Multidisciplinary Optimization, Volume 60 (2019) no. 6, p. 2493 | DOI:10.1007/s00158-019-02337-0
Mehrdad Raisee; Dinesh Kumar; Chris Lacor Non-intrusive Uncertainty Quantification by Combination of Reduced Basis Method and Regression-based Polynomial Chaos Expansion, Uncertainty Management for Robust Industrial Design in Aeronautics, Volume 140 (2019), p. 169 | DOI:10.1007/978-3-319-77767-2_10
Chris Lacor; Éric Savin Polynomial Chaos and Collocation Methods and Their Range of Applicability, Uncertainty Management for Robust Industrial Design in Aeronautics, Volume 140 (2019), p. 687 | DOI:10.1007/978-3-319-77767-2_42
Lukáš Novák; Drahomír Novák, Volume 1978 (2018), p. 470025 | DOI:10.1063/1.5044095
Biswarup Bhattacharyya A critical appraisal of design of experiments for uncertainty quantification, Archives of Computational Methods in Engineering, Volume 25 (2018) no. 3, pp. 727-751 | DOI:10.1007/s11831-017-9211-x | Zbl:1397.62271
Muhannad Aldosary; Jinsheng Wang; Chenfeng Li Structural reliability and stochastic finite element methods, Engineering Computations, Volume 35 (2018) no. 6, p. 2165 | DOI:10.1108/ec-04-2018-0157
Xiangfeng Guo; Daniel Dias; Claudio Carvajal; Laurent Peyras; Pierre Breul Reliability analysis of embankment dam sliding stability using the sparse polynomial chaos expansion, Engineering Structures, Volume 174 (2018), p. 295 | DOI:10.1016/j.engstruct.2018.07.053
Alexander E. David; Giovanni Sansavini Identification of critical states in power systems by limit state surface reconstruction, International Journal of Electrical Power Energy Systems, Volume 101 (2018), p. 162 | DOI:10.1016/j.ijepes.2018.03.004
Lijian Jiang; Qiuqi Li Model reduction method using variable-separation for stochastic saddle point problems, Journal of Computational Physics, Volume 354 (2018), pp. 43-66 | DOI:10.1016/j.jcp.2017.10.056 | Zbl:1380.35170
Panagiotis A. Tsilifis Gradient-Informed Basis Adaptation for Legendre Chaos Expansions, Journal of Verification, Validation and Uncertainty Quantification, Volume 3 (2018) no. 1 | DOI:10.1115/1.4040802
Vahid Yaghoubi; Sadegh Rahrovani; Hassan Nahvi; Stefano Marelli Reduced order surrogate modeling technique for linear dynamic systems, Mechanical Systems and Signal Processing, Volume 111 (2018), p. 172 | DOI:10.1016/j.ymssp.2018.02.020
Lilli Bergner; Christian Kirches The polynomial chaos approach for reachable set propagation with application to chance-constrained nonlinear optimal control under parametric uncertainties, Optimal Control Applications Methods, Volume 39 (2018) no. 2, pp. 471-488 | DOI:10.1002/oca.2329 | Zbl:1393.93020
Nicole Gaus; Carsten Proppe; Cédric Zaccardi Modeling of dynamical systems with friction between randomly rough surfaces, Probabilistic Engineering Mechanics, Volume 54 (2018), p. 82 | DOI:10.1016/j.probengmech.2017.07.004
Sergey Oladyshkin; Wolfgang Nowak Incomplete statistical information limits the utility of high-order polynomial chaos expansions, Reliability Engineering System Safety, Volume 169 (2018), p. 137 | DOI:10.1016/j.ress.2017.08.010
Amit Kumar Rathi; Arunasis Chakraborty Dimension adaptive finite difference decomposition using multiple sparse grids for stochastic computation, Structural Safety, Volume 75 (2018), p. 119 | DOI:10.1016/j.strusafe.2018.06.004
R. Schöbi; B. Sudret; S. Marelli Rare Event Estimation Using Polynomial-Chaos Kriging, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, Volume 3 (2017) no. 2 | DOI:10.1061/ajrua6.0000870
Loïc Giraldi; Olivier P. Le Maître; Kyle T. Mandli; Clint N. Dawson; Ibrahim Hoteit; Omar M. Knio Bayesian inference of earthquake parameters from buoy data using a polynomial chaos-based surrogate, Computational Geosciences, Volume 21 (2017) no. 4, pp. 683-699 | DOI:10.1007/s10596-017-9646-z | Zbl:1369.86007
Qian Shao; Anis Younes; Marwan Fahs; Thierry A. Mara Bayesian sparse polynomial chaos expansion for global sensitivity analysis, Computer Methods in Applied Mechanics and Engineering, Volume 318 (2017), pp. 474-496 | DOI:10.1016/j.cma.2017.01.033 | Zbl:1439.62088
Loïc Le Gratiet; Stefano Marelli; Bruno Sudret Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes, Handbook of Uncertainty Quantification (2017), p. 1289 | DOI:10.1007/978-3-319-12385-1_38
K. Dammak; A. El Hami; S. Koubaa; L. Walha; M. Haddar Reliability based design optimization of coupled acoustic-structure system using generalized polynomial chaos, International Journal of Mechanical Sciences, Volume 134 (2017), p. 75 | DOI:10.1016/j.ijmecsci.2017.10.003
Simon Abraham; Mehrdad Raisee; Ghader Ghorbaniasl; Francesco Contino; Chris Lacor A robust and efficient stepwise regression method for building sparse polynomial chaos expansions, Journal of Computational Physics, Volume 332 (2017), pp. 461-474 | DOI:10.1016/j.jcp.2016.12.015 | Zbl:1384.62216
Alejandra Camacho; Alvaro Talavera; Alexandre A. Emerick; Marco A.C. Pacheco; João Zanni Uncertainty quantification in reservoir simulation models with polynomial chaos expansions: Smolyak quadrature and regression method approach, Journal of Petroleum Science and Engineering, Volume 153 (2017), p. 203 | DOI:10.1016/j.petrol.2017.03.046
Sahand Sabet; Mohammad Poursina Computed torque control of fully-actuated nondeterministic multibody systems, Multibody System Dynamics, Volume 41 (2017) no. 4, pp. 347-365 | DOI:10.1007/s11044-017-9577-4 | Zbl:1418.70013
Vahid Yaghoubi; Stefano Marelli; Bruno Sudret; Thomas Abrahamsson Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation, Probabilistic Engineering Mechanics, Volume 48 (2017), p. 39 | DOI:10.1016/j.probengmech.2017.04.003
Qiuqi Li; Lijian Jiang A novel variable-separation method based on sparse and low rank representation for stochastic partial differential equations, SIAM Journal on Scientific Computing, Volume 39 (2017) no. 6, p. a2879-a2910 | DOI:10.1137/16m1100010 | Zbl:1379.65004
Zequn Wang Piecewise point classification for uncertainty propagation with nonlinear limit states, Structural and Multidisciplinary Optimization, Volume 56 (2017) no. 2, p. 285 | DOI:10.1007/s00158-017-1664-x
Xinfeng Gao; Yijun Wang; Nathan Spotts; Nelson Xie; Sourajeet Roy; Aditi Prasad, 52nd AIAA/SAE/ASEE Joint Propulsion Conference (2016) | DOI:10.2514/6.2016-5057
José David Arregui-Mena; Lee Margetts; Paul M. Mummery Practical application of the stochastic finite element method, Archives of Computational Methods in Engineering, Volume 23 (2016) no. 1, pp. 171-190 | DOI:10.1007/s11831-014-9139-3 | Zbl:1348.65160
Gouthami Senthamaraikkannan; Ian Gates; Vinay Prasad Multiphase reactive-transport simulations for estimation and robust optimization of the field scale production of microbially enhanced coalbed methane, Chemical Engineering Science, Volume 149 (2016), p. 63 | DOI:10.1016/j.ces.2016.04.017
Jérémy Lebon; Guénhaël Le Quilliec; Piotr Breitkopf; Rajan Filomeno Coelho; Pierre Villon Fat Latin Hypercube Sampling and Efficient Sparse Polynomial Chaos Expansion for Uncertainty Propagation on Finite Precision Models: Application to 2D Deep Drawing Process, Computational Methods for Solids and Fluids, Volume 41 (2016), p. 185 | DOI:10.1007/978-3-319-27996-1_8
Dinesh Kumar; Mehrdad Raisee; Chris Lacor An efficient non-intrusive reduced basis model for high dimensional stochastic problems in CFD, Computers and Fluids, Volume 138 (2016), pp. 67-82 | DOI:10.1016/j.compfluid.2016.08.015 | Zbl:1390.76800
Vinzenz Gregor Eck; Wouter Paulus Donders; Jacob Sturdy; Jonathan Feinberg; Tammo Delhaas; Leif Rune Hellevik; Wouter Huberts A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications, International Journal for Numerical Methods in Biomedical Engineering, Volume 32 (2016) no. 8 | DOI:10.1002/cnm.2755
Joseph B. Nagel; Bruno Sudret Spectral likelihood expansions for Bayesian inference, Journal of Computational Physics, Volume 309 (2016), pp. 267-294 | DOI:10.1016/j.jcp.2015.12.047 | Zbl:1351.62077
Katerina Konakli; Bruno Sudret Polynomial meta-models with canonical low-rank approximations: numerical insights and comparison to sparse polynomial chaos expansions, Journal of Computational Physics, Volume 321 (2016), pp. 1144-1169 | DOI:10.1016/j.jcp.2016.06.005 | Zbl:1349.60056
Professor Slawomir Wiak; Peter Offermann; Kay Hameyer Analysis of FE, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Volume 34 (2015) no. 2, p. 596 | DOI:10.1108/compel-07-2014-0174
Danny Weiss; Zohar Yosibash Uncertainty quantification for a 1D thermo-hyperelastic coupled problem using polynomial chaos projection and $p$ -FEMs, Computers Mathematics with Applications, Volume 70 (2015) no. 7, pp. 1701-1720 | DOI:10.1016/j.camwa.2015.04.024 | Zbl:1443.65377
Loïc Le Gratiet; Stefano Marelli; Bruno Sudret Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes, Handbook of Uncertainty Quantification (2015), p. 1 | DOI:10.1007/978-3-319-11259-6_38-1
Konstantin Weise; Luca Di Rienzo; Hartmut Brauer; Jens Haueisen; Hannes Toepfer Uncertainty Analysis in Transcranial Magnetic Stimulation Using Nonintrusive Polynomial Chaos Expansion, IEEE Transactions on Magnetics, Volume 51 (2015) no. 7, p. 1 | DOI:10.1109/tmag.2015.2390593
Ilaria Liorni; Marta Parazzini; Serena Fiocchi; Paolo Ravazzani Study of the Influence of the Orientation of a 50-Hz Magnetic Field on Fetal Exposure Using Polynomial Chaos Decomposition, International Journal of Environmental Research and Public Health, Volume 12 (2015) no. 6, p. 5934 | DOI:10.3390/ijerph120605934
Pierric Kersaudy; Bruno Sudret; Nadège Varsier; Odile Picon; Joe Wiart A new surrogate modeling technique combining Kriging and polynomial chaos expansions - application to uncertainty analysis in computational dosimetry, Journal of Computational Physics, Volume 286 (2015), pp. 103-117 | DOI:10.1016/j.jcp.2015.01.034 | Zbl:1351.94003
D. H. Mac; Z. Tang; S. Clénet; E. Creusé Residual-based a posteriori error estimation for stochastic magnetostatic problems, Journal of Computational and Applied Mathematics, Volume 289 (2015), pp. 51-67 | DOI:10.1016/j.cam.2015.03.027 | Zbl:1325.78011
E. Jacquelin; S. Adhikari; J.-J. Sinou; M.I. Friswell Polynomial chaos expansion in structural dynamics: Accelerating the convergence of the first two statistical moment sequences, Journal of Sound and Vibration, Volume 356 (2015), p. 144 | DOI:10.1016/j.jsv.2015.06.039
Mass Per Pettersson; Gianluca Iaccarino; Jan Nordström Polynomial Chaos Methods, Polynomial Chaos Methods for Hyperbolic Partial Differential Equations (2015), p. 23 | DOI:10.1007/978-3-319-10714-1_3
F. Anstett-Collin; J. Goffart; T. Mara; L. Denis-Vidal Sensitivity analysis of complex models: Coping with dynamic and static inputs, Reliability Engineering System Safety, Volume 134 (2015), p. 268 | DOI:10.1016/j.ress.2014.08.010
Faidra Stavropoulou; Johannes Müller Parametrization of random vectors in polynomial chaos expansions via optimal transportation, SIAM Journal on Scientific Computing, Volume 37 (2015) no. 6, p. a2535-a2557 | DOI:10.1137/130949063 | Zbl:1351.60089
Zhen Gao; Tao Zhou On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification, Communications in Computational Physics, Volume 16 (2014) no. 2, p. 365 | DOI:10.4208/cicp.130813.060214a
Vaibhav Yadav; Sharif Rahman Adaptive-sparse polynomial dimensional decomposition methods for high-dimensional stochastic computing, Computer Methods in Applied Mechanics and Engineering, Volume 274 (2014), pp. 56-83 | DOI:10.1016/j.cma.2014.01.027 | Zbl:1296.62115
G. Migliorati; F. Nobile; E. von Schwerin; R. Tempone Analysis of discrete $L^{2}$ projection on polynomial spaces with random evaluations, Foundations of Computational Mathematics, Volume 14 (2014) no. 3, pp. 419-456 | DOI:10.1007/s10208-013-9186-4 | Zbl:1301.41005
Pierric Kersaudy; Shermila Mostarshedi; Bruno Sudret; Odile Picon; Joe Wiart Stochastic Analysis of Scattered Field by Building Facades Using Polynomial Chaos, IEEE Transactions on Antennas and Propagation, Volume 62 (2014) no. 12, p. 6382 | DOI:10.1109/tap.2014.2359478
Torben Pätz; Tobias Preusser Segmentation of stochastic images using level set propagation with uncertain speed, Journal of Mathematical Imaging and Vision, Volume 48 (2014) no. 3, pp. 467-487 | DOI:10.1007/s10851-013-0421-z | Zbl:1365.68425
Tao Zhou; Akil Narayan; Zhiqiang Xu Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid, SIAM Journal on Scientific Computing, Volume 36 (2014) no. 5, p. A2401 | DOI:10.1137/130950434
Floriane Anstett-Collin; Thierry Mara; Michel Basset Application of global sensitivity analysis to a tire model with correlated inputs, Simulation Modelling Practice and Theory, Volume 44 (2014), p. 54 | DOI:10.1016/j.simpat.2014.03.003
V. Dubourg; B. Sudret Meta-model-based importance sampling for reliability sensitivity analysis, Structural Safety, Volume 49 (2014), p. 27 | DOI:10.1016/j.strusafe.2013.08.010
F. Rupin; G. Blatman; S. Lacaze; T. Fouquet; B. Chassignole Probabilistic approaches to compute uncertainty intervals and sensitivity factors of ultrasonic simulations of a weld inspection, Ultrasonics, Volume 54 (2014) no. 4, p. 1037 | DOI:10.1016/j.ultras.2013.12.006
Bruno Sudret; Géraud Blatman; Marc Berveiller Response Surfaces based on Polynomial Chaos Expansions, Construction Reliability (2013), p. 147 | DOI:10.1002/9781118601099.ch8
Tamara Al‐Bittar; Abdul‐Hamid Soubra Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion, International Journal for Numerical and Analytical Methods in Geomechanics, Volume 37 (2013) no. 13, p. 2039 | DOI:10.1002/nag.2120
C. Soize Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances, Journal of Sound and Vibration, Volume 332 (2013) no. 10, p. 2379 | DOI:10.1016/j.jsv.2011.10.010
N. Roussouly; F. Petitjean; M. Salaun A new adaptive response surface method for reliability analysis, Probabilistic Engineering Mechanics, Volume 32 (2013), p. 103 | DOI:10.1016/j.probengmech.2012.10.001
André Teófilo Beck; Wellison José de Santana Gomes Stochastic fracture mechanics using polynomial chaos, Probabilistic Engineering Mechanics, Volume 34 (2013), p. 26 | DOI:10.1016/j.probengmech.2013.04.002
G. Migliorati; F. Nobile; E. von Schwerin; R. Tempone Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete $L^{2}$ Projection on Polynomial Spaces, SIAM Journal on Scientific Computing, Volume 35 (2013) no. 3, p. A1440 | DOI:10.1137/120897109
Torben Patz; Tobias Preusser Segmentation of Stochastic Images With a Stochastic Random Walker Method, IEEE Transactions on Image Processing, Volume 21 (2012) no. 5, p. 2424 | DOI:10.1109/tip.2012.2187531
Hermann G. Matthies; Elmar Zander Solving stochastic systems with low-rank tensor compression, Linear Algebra and its Applications, Volume 436 (2012) no. 10, pp. 3819-3838 | DOI:10.1016/j.laa.2011.04.017 | Zbl:1241.65016
Christian Soize Stochastic Models of Uncertainties in Computational Structural Dynamics and Structural Acoustics, Nondeterministic Mechanics (2012), p. 61 | DOI:10.1007/978-3-7091-1306-6_2
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Marc Berveiller; Yann Le Pape; Bruno Sudret; Frédéric Perrin Updating the long-term creep strains in concrete containment vessels by using Markov chain Monte Carlo simulation and polynomial chaos expansions, Structure and Infrastructure Engineering, Volume 8 (2012) no. 5, p. 425 | DOI:10.1080/15732479.2010.539057
H. M. Panayirci; G. I. Schuëller On the capabilities of the polynomial chaos expansion method within SFE analysis – an overview, Archives of Computational Methods in Engineering, Volume 18 (2011) no. 1, pp. 43-55 | DOI:10.1007/s11831-011-9058-5 | Zbl:1284.74129
Hermann G. Matthies; Elmar Zander Sparse representations in stochastic mechanics, Computational methods in stochastic dynamics. Selected papers based on the presentations at the 2nd international conference on computational methods in structural dynamics and earthquake engineering (COMPDYN 2009), and 2nd south east European conference on computational mechanics (SEECCM 2009, Rhodos, Greece, June 22–24, 2009 ., New York, NY: Springer, 2011, pp. 247-265 | DOI:10.1007/978-90-481-9987-7_13 | Zbl:1250.82003
Rajan Filomeno Coelho; Philippe Bouillard Multi-Objective Reliability-Based Optimization with Stochastic Metamodels, Evolutionary Computation, Volume 19 (2011) no. 4, p. 525 | DOI:10.1162/evco_a_00034
F. A. Diazdelao; S. Adhikari Gaussian process emulators for the stochastic finite element method, International Journal for Numerical Methods in Engineering, Volume 87 (2011) no. 6, pp. 521-540 | DOI:10.1002/nme.3116 | Zbl:1242.74107
Géraud Blatman; Bruno Sudret Adaptive sparse polynomial chaos expansion based on least angle regression, Journal of Computational Physics, Volume 230 (2011) no. 6, pp. 2345-2367 | DOI:10.1016/j.jcp.2010.12.021 | Zbl:1210.65019
Chao Hu; Byeng D. Youn Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems, Structural and Multidisciplinary Optimization, Volume 43 (2011) no. 3, pp. 419-442 | DOI:10.1007/s00158-010-0568-9 | Zbl:1274.74271
Rajan Filomeno Coelho; Jérémy Lebon; Philippe Bouillard Hierarchical stochastic metamodels based on moving least squares and polynomial chaos expansion: application to the multiobjective reliability-based optimization of space truss structures, Structural and Multidisciplinary Optimization, Volume 43 (2011) no. 5, pp. 707-729 | DOI:10.1007/s00158-010-0608-5 | Zbl:1274.74267
Vincent Dubourg; Bruno Sudret; Jean-Marc Bourinet Reliability-based design optimization using kriging surrogates and subset simulation, Structural and Multidisciplinary Optimization, Volume 44 (2011) no. 5, p. 673 | DOI:10.1007/s00158-011-0653-8
Anthony Nouy Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems, Archives of Computational Methods in Engineering, Volume 17 (2010) no. 4, pp. 403-434 | DOI:10.1007/s11831-010-9054-1 | Zbl:1269.76079
Alban Notin; Nicolas Gayton; Jean Luc Dulong; Maurice Lemaire; Pierre Villon; Haidar Jaffal RPCM: a strategy to perform reliability analysis using polynomial chaos and resampling, European Journal of Computational Mechanics, Volume 19 (2010) no. 8, p. 795 | DOI:10.3166/ejcm.19.795-830
A. Nouy; A. Clément eXtended stochastic finite element method for the numerical simulation of heterogeneous materials with random material interfaces, International Journal for Numerical Methods in Engineering, Volume 83 (2010) no. 10, pp. 1312-1344 | DOI:10.1002/nme.2865 | Zbl:1202.74182
Géraud Blatman; Bruno Sudret An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, Volume 25 (2010) no. 2, p. 183 | DOI:10.1016/j.probengmech.2009.10.003
Géraud Blatman; Bruno Sudret Efficient computation of global sensitivity indices using sparse polynomial chaos expansions, Reliability Engineering System Safety, Volume 95 (2010) no. 11, p. 1216 | DOI:10.1016/j.ress.2010.06.015
Liangsheng Shi; Dongxiao Zhang; Lin Lin; Jinzhong Yang A multiscale probabilistic collocation method for subsurface flow in heterogeneous media, Water Resources Research, Volume 46 (2010) no. 11 | DOI:10.1029/2010wr009066
Anthony Nouy Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Archives of Computational Methods in Engineering, Volume 16 (2009) no. 3, pp. 251-285 | DOI:10.1007/s11831-009-9034-5 | Zbl:1360.65036
Hermann G. Matthies Stochastic finite elements: Computational approaches to stochastic partial differential equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, Volume 88 (2008) no. 11, pp. 849-873 | DOI:10.1002/zamm.200800095 | Zbl:1158.65009

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