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26
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
, 2008
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EFFICIENT UNCERTAINTY QUANTIFICATION USING A TWOSTEP APPROACH WITH CHAOS COLLOCATION
, 2006
"... Abstract. In this paper a Two Step approach with Chaos Collocation for ecient uncertainty quantication in computational
uidstructure interactions is followed. In Step I, a Sensitivity Analysis is used to eciently narrow the problem down from multiple uncertain parameters to one parameter which ha ..."
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Abstract. In this paper a Two Step approach with Chaos Collocation for ecient uncertainty quantication in computational
uidstructure interactions is followed. In Step I, a Sensitivity Analysis is used to eciently narrow the problem down from multiple uncertain parameters to one parameter which has the largest in
uence on the solution. In Step II, for this most important parameter the Chaos Collocation method is employed to obtain the stochastic response of the solution. The Chaos Collocation method is presented in this paper, since a previous study showed that no ecient method was available for arbitrary probability distributions. The Chaos Collocation method is compared on eciency with Monte Carlo simulation, the Polynomial Chaos method, and the Stochastic Collocation method. The Chaos Collocation method is nonintrusive and shows exponential convergence with respect to the polynomial order for arbitrary parameter distributions. Finally, the eciency of the Two Step approach with Chaos Collocation is demonstrated for the linear piston problem with an unsteady boundary condition. A speedup of a factor of 100 is obtained compared to a full uncertainty analysis for all parameters. 1
Uncertainty quantification in hybrid dynamical systems
 J. Comput. Phys
, 2013
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Uncertainty quantification and propagation in structural dynamics
 International Conference on Civil Engineering in the New Millennium: Opportunities and Challenges, Howrah
, 2007
"... In many stochastic mechanics problems the solution of a system of coupled linear random algebraic equations is needed. This problem in turn requires the calculation of the inverse of a random matrix. Over the past four decades several approximate analytical methods and simulation methods have been p ..."
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In many stochastic mechanics problems the solution of a system of coupled linear random algebraic equations is needed. This problem in turn requires the calculation of the inverse of a random matrix. Over the past four decades several approximate analytical methods and simulation methods have been proposed for the solution of this problem in the context of probabilistic structural mechanics. In this paper, for the first time, we present an exact analytical method for the inverse of a real symmetric (in general nonGaussian) random matrix of arbitrary dimension. The proposed method is based on random matrix theory and utilizes the Jacobian of the underlying nonlinear matrix transformation. Exact expressions for the mean and covariance of the response vector is obtained in closedform. Numerical examples are given to illustrate the use of the expressions derived in the paper.
POLYNOMIAL CHAOS EXPANSION FOR GENERAL MULTIVARIATE DISTRIBUTIONS WITH CORRELATED VARIABLES
"... Abstract. Recently, the use of Polynomial Chaos Expansion (PCE) has been increasing to study the uncertainty in mathematical models for a wide range of applications and several extensions of the original PCE technique have been developed to deal with some of its limitations. But as of to date PCE ..."
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Abstract. Recently, the use of Polynomial Chaos Expansion (PCE) has been increasing to study the uncertainty in mathematical models for a wide range of applications and several extensions of the original PCE technique have been developed to deal with some of its limitations. But as of to date PCE methods still have the restriction that the random variables have to be statistically independent. This paper presents a method to construct a basis of the probability space of orthogonal polynomials for general multivariate distributions with correlations between the random input variables. We show that, as for the current PCE methods, the statistics like mean, variance and Sobol ’ indices can be obtained at no significant extra postprocessing costs. We study the behavior of the proposed method for a range of correlation coefficients for an ODE with model parameters that follow a bivariate normal distribution. In all cases the convergence rate of the proposed method is analogous to that for the independent case. Finally, we show, for a canonical enzymatic reaction, how to propagate experimental errors through the process of fitting parameters to a probabilistic distribution of the quantities of interest, and we demonstrate the significant difference in the results assuming independence or full correlation compared to taking into account the true correlation.
Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems,”
, 2011
"... This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external for ..."
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This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all reallife systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work, uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a leastsquare collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraintbased multiobjective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or offset, the resulting Pareto optimal tradeoff curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.
Differential Equations
, 2013
"... This thesis is about preconditioning techniques for time dependent stochastic Partial Differential Equations arising in the broader context of Uncertainty Quantification. Stateoftheart methods for an efficient integration of stochastic PDEs require the solution field to lie on a low dimensional l ..."
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This thesis is about preconditioning techniques for time dependent stochastic Partial Differential Equations arising in the broader context of Uncertainty Quantification. Stateoftheart methods for an efficient integration of stochastic PDEs require the solution field to lie on a low dimensional linear manifold. In cases when there is not such an intrinsic low rank structure we must resort on expensive and time consuming simulations. We provide a preconditioning technique based on local time stretching capable to either push or keep the solution field on a low rank manifold with substantial reduction in the storage and the computational burden. As a byproduct we end up addressing also classical issues related to long time integration of stochastic PDEs.
A NOVEL GALERKIN PROJECTION APPROACH FOR DAMPED STOCHASTIC DYNAMIC SYSTEMS
"... Abstract. This article provides the theoretical development and simulation results of a novel ..."
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Abstract. This article provides the theoretical development and simulation results of a novel
Computer Science Technical Report "Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems" Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems
"... Abstract This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal ..."
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Abstract This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.