Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

By Christian Soize, Roger G. Ghanem. SIAM Journal on Scientific Computing, 2005

634 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/9572ac6926c03a7c2557c90c83d5968835aabab8

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Citations in the corpus

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, background): “For SP , classical choices consists of polynomial spaces [28, 29, 9], piecewise polynomial spaces [30, 31, 32] or more general orthogonal bases [33].”
Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory‐scale porous media (2011, background): “[18] We start by noting that any square integral random response admits the following chaos representation [Soize and Ghanem, 2004], f ðfÞ ¼ X NM s ðfÞ ¼ so 0 þ XM i¼1 si 1ð iÞ þ XM j i sij 2ð i; jÞ þ XM k j sijk 3ð i; j; kÞ þ . . . ð5Þ where is a th-order multivariate orthogonal polynomial, f is…”
A compressed sensing approach for partial differential equations with random input data (2012, background): “In [42], a generalization to bases spanned by functionals associated with random variables of arbitrary measure was proposed.”
Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos (2013, background): “An arbitrary polynomial basis may be defined to reuse the Cartesian coordinate samples [61], but this yields a set of orthogonal polynomials unique to the initial conditions and their uncertainty.”
Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter (2009, background): “Typically, nonlinear functionals of x are expanded with respect to a basis in the space [Soize and Ghanem, 2004] of square-integrable random variables.”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, background): “In this context, it leads to a representation of functionals of random variables alternative to classical chaos expansions [28, 12, 27, 26, 29].”
Global sensitivity analysis: a flexible and efficient framework with an example from stochastic hydrogeology (2012, methodology): “To accommodate for a wide range of data distributions, a recent generalization of PCE denoted as the arbitrary polynomial chaos (aPC) was developed [29–33].”
Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites (2008, methodology): “(where δ is the Kronecker delta) and has a probability distribution which depends on the probability law of the random eld f and which can be constructed as explained in [11].”
Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression (2014, background): “For instance, when Ξi follows a uniform or Gaussian distribution, ψki(Ξi) are (normalized) Legendre or Hermite polynomials, respectively, [5, 6].”
Polynomial-Chaos-based Kriging (2015, positive): “, E [ Y 2 ] < +∞), it can be cast as the following polynomial chaos expansion (PCE) [6, 51]: Y ≡M(X) = ∑ α∈NM aα ψα(X), (2)”
Adaptive sparse polynomial chaos expansion based on least angle regression (2011, background): “This random output may be represented explicitly in a suitable space spanned by the polynomial chaos (PC) basis [1] [2] which is a series of multivariate polynomials that are orthogonal with respect to the joint distribution of the input variables.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “The reader can refer to [43] for a general methodology for the construction of approximation spaces SiPi in the case of arbitrary probability measures Pξi .”
Bayesian inference with optimal maps (2011, background): “That said, orthogonal polynomial expansions for dependent random variables with arbitrary probability measure can certainly be constructed [51].”