Polynomial chaos expansions and stochastic finite element methods

This paper is a state-of-the art review on sparse polynomial chaos expansions (PCE) for engineering applications. It contains a step-by-step presentation of PCEs and their use in moment-, sensitivity- and reliability analysis. Error estimators and sparse expansions for addressing high-dimensional problems are discussed. Applications in geotechnical engineering showcase the efficiency of sparse PCEs.

[1]  Seymour Geisser,et al.  The Predictive Sample Reuse Method with Applications , 1975 .

[2]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[3]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[4]  Bak Kong Low,et al.  Reliability analysis of circular tunnel under hydrostatic stress field , 2010 .

[5]  M. Berveiller.  PRESENTATION OF TWO METHODS FOR COMPUTING THE RESPONSE COEFFICIENTS IN STOCHASTIC FINITE ELEMENT ANALYSIS M. Berveiller, and B. Sudret Electricité de France, R&D Division, Site des Renardières F-77818 Moret sur Loing , 2004 .

[6]  William H. Press,et al.  Numerical recipes , 1990 .

[7]  K. Phoon,et al.  Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme , 2002 .

[8]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[9]  S. Isukapalli.  UNCERTAINTY ANALYSIS OF TRANSPORT-TRANSFORMATION MODELS , 1999 .

[10]  G. Blatman,et al.  Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis , 2009 .

[11]  S. L. Lee,et al.  Reliability Analysis of Pile Settlement , 1990 .

[12]  K. Phoon,et al.  Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion , 2005 .

[13]  Gregory B. Baecher,et al.  Estimating Autocovariance of In‐Situ Soil Properties , 1993 .

[14]  Kok-Kwang Phoon,et al.  Simulation of second-order processes using Karhunen–Loeve expansion , 2002 .

[15]  Henrik O. Madsen,et al.  Structural Reliability Methods , 1996 .

[16]  Bruno Sudret,et al.  Meta-models for structural reliability and uncertainty quantification , 2012, 1203.2062.

[17]  G. Fenton.  Random Field Modeling of CPT Data , 2000 .

[18]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[19]  Bruno Sudret,et al.  Sparse polynomial chaos expansions of vector-valued response quantities , 2014 .

[20]  I. Sobol.  Global Sensitivity Indices for Nonlinear Mathematical Models , 2004 .

[21]  Hans-Jürgen Lang,et al.  Bodenmechanik und Grundbau , 1982 .

[22]  Erik H. Vanmarcke,et al.  Random Fields: Analysis and Synthesis. , 1985 .

[23]  Comparison of methods for computing the response coefficients in stochastic finite element analysis , 2003 .

[24]  Yoshua Bengio,et al.  Model Selection for Small Sample Regression , 2002, Machine Learning.

[25]  Nut Mao,et al.  Probabilistic analysis and design of strip foundations resting on rocks obeying Hoek-Brown failure criterion , 2012 .

[26]  Tamara Al-Bittar,et al.  Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion , 2011 .

[27]  B. Sudret,et al.  An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis , 2010 .

[28]  George Deodatis,et al.  Effects of random heterogeneity of soil properties on bearing capacity , 2005 .

[29]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[30]  K. Phoon,et al.  Comparison between Karhunen-Loève expansion and translation-based simulation of non-Gaussian processes , 2007 .

[31]  R. Nelsen.  An Introduction to Copulas , 1998 .

[32]  Joseph A. C. Delaney.  Sensitivity analysis , 2018, The African Continental Free Trade Area: Economic and Distributional Effects.

[33]  Gordon A. Fenton,et al.  Estimation for Stochastic Soil Models , 1999 .

[34]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[35]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[36]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[37]  Etienne de Rocquigny,et al.  Modelling Under Risk and Uncertainty: An Introduction to Statistical, Phenomenological and Computational Methods , 2012 .

[38]  Bruno Sudret,et al.  Principal component analysis and Least Angle Regression in spectral stochastic finite element analysis , 2011 .

[39]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[40]  W. Tang,et al.  Efficient Spreadsheet Algorithm for First-Order Reliability Method , 2007 .

[41]  R. Tibshirani,et al.  Forward stagewise regression and the monotone lasso , 2007, 0705.0269.

[42]  E. Vanmarcke.  Probabilistic Modeling of Soil Profiles , 1977 .

[43]  Bruno Sudret,et al.  Stochastic finite element methods in geotechnical engineering , 2008 .

[44]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[45]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[46]  Dirk P. Kroese,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[47]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[48]  I. Sobol,et al.  Global sensitivity indices for nonlinear mathematical models. Review , 2005 .

[49]  Bak Kong Low,et al.  Reliability-based design applied to retaining walls , 2005 .

[50]  M. Lemaire,et al.  Stochastic finite element: a non intrusive approach by regression , 2006 .

[51]  Stefano Marelli,et al.  UQLab: a framework for uncertainty quantification in MATLAB , 2014 .

[52]  Bruno Sudret,et al.  Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach , 2008 .

[53]  Saltelli Andrea,et al.  Global Sensitivity Analysis: The Primer , 2008 .

[54]  Bruno Sudret,et al.  A stochastic finite element procedure for moment and reliability analysis , 2006 .