Efficient Collocational Approach for Parametric Uncertainty Analysis

By Dongbin Xiu. Communications in Computational Physics, 2007

475 citations recorded by the source corpus.

A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations.

Subfields: General Health Professions, Philosophy

View Paper: https://www.semanticscholar.org/paper/1e5aeb9be873adfce4997e55d0991df780de2946

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Citations in the corpus

Galerkin Method for Wave Equations with Uncertain Coefficients (2008, background): “, [2, 6, 8], its application to hyperbolic problems has been limited as of now.”
Fast Numerical Methods for Stochastic Computations: A Review (2009, methodology): “10) and is termed as “aliasing error” in [86], following the similar terminology from the classical deterministic spectral methods.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “Several methods have been proposed for the de nition and computation of the approximate solution: L(2) projection [20,22], interpolation [2,15,52,51,49], regression [6] or Galerkin projections [19,3,35,17].”
Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification (2013, methodology): “A partial remedy of the situation can be found by using sparse grids (SG) based on the Smolyak algorithm [3], which have a weaker dependence on the dimensionality of the problem (see [4] [5] [6] and the adaptive version developed by our group [7]).”
Probability density evolution analysis of engineering structures via cubature points (2012, methodology): “[41–43] successfully combined the Galerkin projection of polynomial chaos and the sparse grid method.”
Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems (2011, methodology): “The standard UQ methods such as Monte Carlo (MC) (Fishman, 1996) either exhibit poor convergence rates or others such as quasi Monte Carlo (QMC) (Niederreter, 1992; Sloan & Joe, 1994), generalized polynomial chaos (gPC) (Xiu & Karniadakis, 2002) and the associated probabilistic collocation method (P”
Computational Aspects of Stochastic Collocation with Multifidelity Models (2014, unknown): “, [1, 2, 3, 4, 5, 7, 8, 12, 15, 17, 19, 21].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, methodology): “A rst way to solve a PDE de ned on a random domain consists in using direct spectral stochastic approaches (interpolation [8, 9, 10, 11, 12], projection [13, 14, 15], regression [16]), or any other direct stochastic method such as Monte-Carlo or quasi Monte-Carlo simulations [17], coupled with a det”
Stochastic collocation and mixed finite elements for flow in porous media (2008, methodology): “A very promising approach for improving the efficiency of non-sampling methods is the stochastic collocation method [2,39,38].”
Sparse grid collocation schemes for stochastic natural convection problems (2007, background): “Recently, rigorous error estimates for the collocation based stochastic methodology have been developed in [27] and [35].”
Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion (2012, background): “The difference between expans ion in an optimal basis and expansion of Y(ξ ) after transformation to ξN is the so-called “aliasing error ”(see [59]).”
A stochastic collocation approach to Bayesian inference in inverse problems (2009, methodology): “…Q , of an integration rule (cubature) on R n z such that The expansion of (3.8) thus becomes an approximation to the exact expansion (3.1); that is, e The difference between the two expansions is the so-called “aliasing error” [25] and is induced by the error of using the integration rule in (3.9).”
Efficient posterior exploration of a high‐dimensional groundwater model from two‐stage Markov chain Monte Carlo simulation and polynomial chaos expansion (2013, background): “This deviation is also referred to as the ‘‘aliasing’’ error [Xiu, 2007] and decreases with Q in equation (13).”
Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, methodology): “As a comparison, we also use the pseudospectral SCM [40] to construct a Legendre PC approximation to the forward model G(Z).”
A comparative study of uncertainty propagation methods for black-box-type problems (2008, methodology): “Approaches using the Latin Hyper cube sampling (Choi et al. 2004) and the numerical integration ( Xiu 2007 ) have been reported.”
Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed (2010, methodology): “Two of the major approaches of high-order stochastic collocation methods are the Lagrange interpolation approach, see Xiu and Hesthaven [15] and later (independently) in [1], and the pseudo-spectral gPC approach from [13].”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “Other non-intrusive methods are based on least-square regression as discussed in [16, 34, 35, 69], or on pseudo-spectral approximation as discussed in [91, 26].”