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Fast Numerical Methods for Stochastic Computations: A Review
, 2009
"... This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework ..."
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Cited by 65 (2 self)
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This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multidimensional random spaces.
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Report 201003, Seminar for Applied Mathematics
, 2010
"... Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PD ..."
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Cited by 43 (10 self)
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Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space V = H10 (D) of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear Finite Element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates afforded by the best Nterm sequence approximations in the parameter space and the rate of Finite Element approximations in D for a single instance of the parametric problem. 1
Convergence rates of best Nterm Galerkin approximations for a class of elliptic sPDEs ∗
, 2010
"... Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 (D)orthogonal bases, a ..."
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Cited by 36 (9 self)
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Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 (D)orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y(ω) = (yi(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x, y) is a function of both the space variable x ∈ D and the in general countably many parameters y. We establish new regularity theorems decribing the smoothness properties of the solution u as a map from y ∈ U = (−1, 1) ∞ to V = H 1 0(D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a socalled “generalized polynomial chaos”(gpc) expansion of u. Convergence estimates of approximations of u by best Nterm truncated Vvalued polynomials in the variable y ∈ U are established. These estimates are of the form N −r, where the rate of convergence r depends only on the decay of the random input expansion. It
LowRank Tensor Krylov Subspace Methods for Parametrized Linear Systems
, 2010
"... We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1,...,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse ..."
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Cited by 25 (3 self)
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We consider linear systems A(α)x(α) = b(α) depending on possibly many parameters α = (α1,...,αp). Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse of dimensionality can be avoided for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that x(α) can be well approximated by a tensor of low rank. In particular, lowrank tensor variants of shortrecurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
, 2008
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Uncertainty quantification and weak approximation of an elliptic inverse problem
 SIAM J. Numer. Anal
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Spectral methods for parameterized matrix equations
 SIAM. J. Matrix Anal. Appl
"... Abstract. We apply polynomial approximation methods — known in the numerical PDEs context as spectral methods — to approximate the vectorvalued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudo ..."
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Cited by 10 (3 self)
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Abstract. We apply polynomial approximation methods — known in the numerical PDEs context as spectral methods — to approximate the vectorvalued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residualminimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples. Key words. parameterized systems, spectral methods 1. Introduction. We
A leastsquares approximation of highdimensional uncertain systems
 Center for Turbulence Research
, 2007
"... The behavior and evolution of complex systems are known only partially due to lack of knowledge about the governing physical laws or limited information regarding their operating conditions and input parameters (e.g., material properties). Uncertainty quantication (UQ) plays a crucial role in the ..."
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Cited by 9 (2 self)
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The behavior and evolution of complex systems are known only partially due to lack of knowledge about the governing physical laws or limited information regarding their operating conditions and input parameters (e.g., material properties). Uncertainty quantication (UQ) plays a crucial role in the construction of credible mathematical/computational models for such systems. In this context a set of a partial dierential equations (PDEs) are considered. The uncertain parameters are assimilated based on the available information and are described as random variables in a probabilistic framework. A computational model is then dened to approximate the statistics of the solution of these PDEs, thus propagating the uncertainty from the input parameters to the response of the system. Bringing the uncertainty propagation schemes into focus, the Monte Carlo sampling has been utilized for a long time as a general purpose scheme. There has recently been an increasing interest in developing more ecient computational models for the analysis of uncertain systems as compared to Monte Carlo techniques that are generally known to have a slow rate of convergence. In particular, perturbationbased techniques, Kleiber & Hien (1992), are shown to be eective for situations where the input parameters exhibit
Discrete least squares polynomial approximation with random evaluations application to parametric and stochastic elliptic
, 2013
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A Validation Study of a Stochastic Representation of Composite Material Properties from Limited Experimental Data
"... This work aims at characterising some stochastic properties of heterogeneous composite fabric from a limited number of vibration tests. The frequency response functions, measured in these tests, are implemented in a deterministic inverse problem in order to construct a database consisting of spatial ..."
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Cited by 5 (0 self)
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This work aims at characterising some stochastic properties of heterogeneous composite fabric from a limited number of vibration tests. The frequency response functions, measured in these tests, are implemented in a deterministic inverse problem in order to construct a database consisting of spatial estimations of the Young’s modulus of the composite material. The database is then assimilated to construct a stochastic model characterising the Young’s modulus. The predictive model is set up to account for both aleatory uncertainties related to sampletosample variabilities of models parameters and, more importantly, epistemic uncertainties related to the insufficiency of the available data. To this end a probabilistic model based on polynomial chaos expansions is implemented. To represent the uncertainty due to the finitesize ensemble measurements, the coefficients of the chaos expansion are themselves considered as random variables. A Bayesian inference scheme with Markov chain Monte Carlo has been implemented to characterise these coefficients. A cross validation approach has been used to validate the constructed stochastic model taking into consideration the epistemic uncertainty. List of Acronyms (e)FIM (empirical) Fisher information matrix DOF(s) degree(s) of freedom FE finite element FRF(s) frequency response function(s) KL KarhunenLoève MAP a maximum a posteriori probability estimate MCMC Markov chain Monte Carlo