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The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations
- SIAM J. SCI. COMPUT
, 2002
"... We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dime ..."
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Cited by 398 (42 self)
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We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte-Carlo simulations for low dimensional stochastic inputs.
Phase transition of the largest eigenvalue for non-null complex sample covariance matrices
, 2008
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Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition
, 2003
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Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos
, 2002
"... We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations suject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and so ..."
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Cited by 94 (16 self)
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We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations suject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener (1938). A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations.
Advanced determinant calculus: a complement
- LINEAR ALGEBRA APPL
, 2005
"... This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular probl ..."
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Cited by 89 (8 self)
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This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems.
The arctic circle boundary and the Airy process
- Ann. Prob
, 2005
"... Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the struc ..."
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Cited by 88 (6 self)
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Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond. 1. Introduction and
Distribution on partitions, point processes, and the hypergeometric kernel
- COMM. MATH. PHYS
, 1999
"... We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. Th ..."
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Cited by 70 (28 self)
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We study a 3–parametric family of stochastic point processes on the one–dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel. In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed before, see math.RT/9810015, the correlation functions of these limit processes also have determinantal form with so–called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel. The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a ‘discrete integrable operator’. We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel for Meixner orthogonal polynomials. This fact
Orthogonal polynomials for exponential weights x2ρe−2Q(x) on [0,d
- J. Approx. Theory
"... xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + ..."
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Cited by 68 (18 self)
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xi + 476 pp With each of a large class of positive measures µ on the real line it is possible to associate a sequence {pn(x)} of orthogonal polynomials with the property that pm(x)pn(x)dµ(x) = 0, m ̸ = n, 1, m = n. (1) Such a sequence satisfies a three term recurrence relation xpn(x) = anPn+1(x) + bnPn(x) + an−1Pn−1(x) (2) Conversely, for suitable starting values and coefficient sequences {an}, {bn}, the recurrence relation (2) generates a sequence of polynomials satisfying (1) for some measure µ. The polynomials have their zeros within the interval of support of the measure. Examples date from the 19th century. The Jacobi polynomials P (α,β) n (x) are orthogonal with respect to µ with support [−1, 1], where dµ = (1 − x) α+1 (1 +
A new type of limit theorems for the one-dimensional quantum random walk
, 2003
"... In this paper we consider the one-dimensional quantum random walk Xϕ n at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U. We give a combinatorial. The expression clarifies the dependence of it on expression for the characteristic function of Xϕ n components of unit ..."
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Cited by 68 (29 self)
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In this paper we consider the one-dimensional quantum random walk Xϕ n at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U. We give a combinatorial. The expression clarifies the dependence of it on expression for the characteristic function of Xϕ n components of unitary matrix U and initial qubit state ϕ. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that Xϕ n/n converges in distribution to a limit Zϕ as n → ∞ where Zϕ has a density 1/π(1−x 2) √ 1 − 2x2 for x ∈ ( − √ 2/2, √ 2/2). Moreover we discuss some known simulation results based on our limit theorems.
Fluctuation properties of the TASEP with periodic initial configuration
, 2006
"... We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and det ..."
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Cited by 67 (34 self)
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We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [27] and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
