Results 1  10
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102
Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations
, 2009
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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.
Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems
, 2010
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Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
, 2008
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Adaptive Smolyak pseudospectral approximation
 SIAM Journal on Scientific Computing
, 2012
"... Abstract. Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak’s algorithm with generalized sparse grids. We rigorously analyze and extend the ..."
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Cited by 12 (4 self)
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Abstract. Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak’s algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem. Key words. Smolyak algorithms, sparse grids, orthogonal polynomials, pseudospectral approximation, approximation theory, uncertainty quantification AMS subject classifications. 41A10, 41A63, 47A80, 65D15, 65D32 1. Introduction. A
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
Stochastic modeling of uncertainties in computational structural dynamics  Recent theoretical advances
, 2012
"... This paper deals with a short overview on stochastic modeling of uncertainties. We introduce the types of uncertainties, the variability of real systems, the types of probabilistic approaches, the representations for the stochastic models of uncertainties, the construction of the stochastic models u ..."
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Cited by 8 (3 self)
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This paper deals with a short overview on stochastic modeling of uncertainties. We introduce the types of uncertainties, the variability of real systems, the types of probabilistic approaches, the representations for the stochastic models of uncertainties, the construction of the stochastic models using the maximum entropy principle, the propagation of uncertainties, the methods to solve the stochastic dynamical equations, the identification of the prior and the posterior stochastic models, the robust updating of the computational models and the robust design with uncertain computational models. We present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties, and for their identification with experimental data. We also present the construction of the posterior stochastic model of uncertainties using the Bayesian method when experimental data are available.
Uncertainty quantification for integrated circuits: Stochastic spectral methods
 in Proc. Int. Conf. ComputerAided Design
, 2013
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Stochastic Optimal Robin Boundary Control Problems of AdvectionDominated Elliptic Equations
"... Abstract. In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advectiondffusionreaction elliptic equation with advectiondominated term. We assume that the uncertainty comes from the advection filed and consider a stochastic Robin boundary condition as c ..."
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Cited by 4 (2 self)
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Abstract. In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advectiondffusionreaction elliptic equation with advectiondominated term. We assume that the uncertainty comes from the advection filed and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided.