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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 59 (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.
Phase Change of Limit Laws in the Quicksort Recurrence Under Varying Toll Functions
, 2001
"... We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "to ..."
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Cited by 54 (18 self)
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We characterize all limit laws of the quicksort type random variables defined recursively by Xn = X In + X # n1In + Tn when the "toll function" Tn varies and satisfies general conditions, where (Xn ), (X # n ), (I n , Tn ) are independent, Xn . . . , n 1}. When the "toll function" Tn (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n## log E(Tn )/ log n 1/2), Xn is asymptotically normally distributed; nonnormal limit laws emerge when Tn becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an insitu permutation algorithm to tree traversal algorithms, etc.
Stochastic modeling of flow structure interactions using generalized polynomial chaos
 JOURNAL OF FLUIDS ENGINEERING
, 2002
"... We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flowstructure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standar ..."
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Cited by 30 (2 self)
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We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flowstructure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener (1938) and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to secondorder oscillators to demonstrate convergence, and subsequently is coupled to incompressible NavierStokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortexinduced vibrations.
Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also de ..."
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Cited by 28 (1 self)
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
MODULAR SYMBOLS HAVE A NORMAL DISTRIBUTION
 GAFA GEOMETRIC AND FUNCTIONAL ANALYSIS
, 2004
"... We prove that the modular symbols appropriately normalized and ordered have a Gaussian distribution for all cofinite subgroups of SL2(R). We use spectral deformations to study the poles and the residues of Eisenstein series twisted by power of modular symbols. ..."
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Cited by 23 (10 self)
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We prove that the modular symbols appropriately normalized and ordered have a Gaussian distribution for all cofinite subgroups of SL2(R). We use spectral deformations to study the poles and the residues of Eisenstein series twisted by power of modular symbols.
Limit theorems for pattern in phylogenetic trees
 Journal of Mathematical Biology
, 2010
"... Studying the shape of phylogenetic trees under different random models is an important issue in evolutionary biology. In this paper, we propose a general framework for deriving detailed statistical results for patterns in phylogenetic trees under the YuleHarding model and the uniform model, two of ..."
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Cited by 14 (8 self)
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Studying the shape of phylogenetic trees under different random models is an important issue in evolutionary biology. In this paper, we propose a general framework for deriving detailed statistical results for patterns in phylogenetic trees under the YuleHarding model and the uniform model, two of the most fundamental random models considered in phylogenetics. Our framework will unify several recent studies which were mainly concerned with the mean value and the variance. Moreover, refined statistical results such as central limit theorems, BerryEsseen bounds, local limit theorems, etc. are obtainable with our approach as well. A key contribution of the current study is that our results are applicable to the whole range of possible sizes of the pattern. 1
BV Estimates for Multicomponent Chromatography with Relaxation
, 1999
"... . We consider the Cauchy problem for a system of 2n balance laws which arises from the modelling of multicomponent chromatography: 8 ? ! ? : u t + u x = \Gamma 1 " \Gamma F (u) \Gamma v \Delta ; v t = 1 " \Gamma F (u) \Gamma v \Delta ; (1) This model describes a liquid flowi ..."
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Cited by 11 (4 self)
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. We consider the Cauchy problem for a system of 2n balance laws which arises from the modelling of multicomponent chromatography: 8 ? ! ? : u t + u x = \Gamma 1 " \Gamma F (u) \Gamma v \Delta ; v t = 1 " \Gamma F (u) \Gamma v \Delta ; (1) This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors u = (u 1 ; : : : ; u n ) and v = (v 1 ; : : : ; v n ). We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times t 0. Moreover, using the L 1 distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t. ". Finally we prove that as " ! 0, the solutions of (1) converge to a limit described by the system \Gamma u + F (u) \Delta t + u x = 0; v = F (u): (2) The proof of the u...
eXtended Stochastic Finite Element Method for the numerical simulation of heterogenous materials with random material interfaces
, 2010
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EFFICIENT STOCHASTIC GALERKIN METHODS FOR RANDOM DIFFUSION EQUATIONS
"... Abstract. We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projec ..."
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Cited by 10 (1 self)
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Abstract. We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projection. Although the resulting system of diffusion equations are coupled, we show that one can construct fast numerical methods to solve them in a decoupled fashion. The methods are based on separation of the diagonal terms and offdiagonal terms in the matrix of the Galerkin system. We examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters. Numerical examples are provided, for both steady and unsteady random diffusions, to support the analysis. Key words. Generalized polynomial chaos, stochastic Galerkin, random diffusion, uncertainty quantification 1. Introduction. We
An Efficient Spectral Method for Acoustic Scattering from Rough Surfaces
, 2007
"... Abstract. An efficient and accurate spectral method is presented for scattering problems with rough surfaces. A probabilistic framework is adopted by modeling the surface roughness as random process. An improved boundary perturbation technique is employed to transform the original Helmholtz equation ..."
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Cited by 8 (2 self)
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Abstract. An efficient and accurate spectral method is presented for scattering problems with rough surfaces. A probabilistic framework is adopted by modeling the surface roughness as random process. An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain. The generalized polynomial chaos (gPC) is then used to discretize the random space; and a FourierLegendre method to discretize the physical space. These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces. Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm.