• Documents
  • Authors
  • Tables
  • Log in
  • Sign up
  • MetaCart
  • DMCA
  • Donate

CiteSeerX logo

Advanced Search Include Citations
Advanced Search Include Citations | Disambiguate

The Wiener-Askey polynomial chaos for stochastic differential equations (0)

by D Xiu, G Karniadakis
Venue:SIAM J. Sci. Comput
Add To MetaCart

Tools

Sorted by:
Results 1 - 10 of 398
Next 10 →

High-Order Collocation Methods for Differential Equations with Random Inputs

by Dongbin Xiu, Jan, S. Hesthaven - SIAM Journal on Scientific Computing
"... Abstract. Recently there has been a growing interest in designing efficient methods for the so-lution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling met ..."
Abstract - Cited by 188 (13 self) - Add to MetaCart
Abstract. Recently there has been a growing interest in designing efficient methods for the so-lution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementa-tions of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a high-order stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods. Key words. collocation methods, stochastic inputs, differential equations, uncertainty quantifi-cation

Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations

by Hermann G. Matthies, Andreas Keese , 2003
"... ..."
Abstract - Cited by 113 (6 self) - Add to MetaCart
Abstract not found

Preconditioning stochastic Galerkin saddle point systems

by Catherine E. Powell, Elisabeth Ullmann - SIAM J. MATRIX ANAL. APPL , 2009
"... Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which ..."
Abstract - Cited by 110 (4 self) - Add to MetaCart
Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.
(Show Context)

Citation Context

...well-posedness of (2.1)–(2.2), primal variational formulations, and approximation schemes based on finite element spatial discretizations have been widely studied (see [9], [4], [5], [3], [13], [23], =-=[35]-=-). Stochastic Galerkin approximation, specifically, has been studied in [4] and [9] and solvers for the resulting symmetric positive definite linear systems have been studied in [27], [32] and [30]. 2...

ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS

by Oliver G. Ernst, Antje Mugler, Hans-jörg Starkloff, Elisabeth Ullmann
"... A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal w ..."
Abstract - Cited by 97 (3 self) - Add to MetaCart
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos

by Dongbin Xiu, George Em Karniadakis , 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 ..."
Abstract - Cited by 94 (16 self) - Add to MetaCart
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.

An adaptive multi-element generalized polynomial chaos method for stochastic differential equations

by Xiaoliang Wan, George Em Karniadakis - J. COMPUT. PHYS , 2005
"... We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other ra ..."
Abstract - Cited by 82 (11 self) - Add to MetaCart
We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the rela-tive error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized Polynomial Chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan-Orszag three-mode problem, as well as advection-diffusion problems. The new method is similar to spectral element method for deterministic problems but with h-p discretization of the random space.
(Show Context)

Citation Context

...gPC)”, was introduced in ∗ Corresponding author. Email address: xlwan@dam.brown.edu and gk@dam.brown.edu (Xiaoliang Wan and George Em Karniadakis). Preprint submitted to Elsevier Science 9 March 2005 =-=[4,5]-=-. This extension includes a family of orthogonal polynomials (the so-called Askey scheme) from which the trial basis is selected, and can represent nonGaussian processes more efficiently; it includes ...

An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data

by Fabio Nobile, Raul Tempone, Clayton G. Webster , 2007
"... ..."
Abstract - Cited by 80 (16 self) - Add to MetaCart
Abstract not found

Fast Numerical Methods for Stochastic Computations: A Review

by Dongbin Xiu , 2009
"... This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework ..."
Abstract - Cited by 76 (5 self) - Add to MetaCart
This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order 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 multi-dimensional random spaces.

Efficient collocational approach for parametric uncertainty analysis

by Dongbin Xiu - Commun. Comput. Phys , 2007
"... Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as co ..."
Abstract - Cited by 65 (6 self) - Add to MetaCart
Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A high-order stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudo-spectral ” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. Key words: Collocation methods; pseudo-spectral methods; stochastic inputs; random differential equations; uncertainty quantification. 1
(Show Context)

Citation Context

...ncur excessive computational burden, especially for systems that are already computationally intensive in their deterministic settings. A recently developed method, generalized polynomial chaos (gPC) =-=[28]-=-, belong to the class of non-sampling methods. With gPC, stochastic quantities are expressed as orthogonal polynomials of the input random parameters, and different types of orthogonal polynomials can...

A generalized spectral decomposition technique to solve a class of linear stochastic . . .

by Anthony Nouy , 2007
"... ..."
Abstract - Cited by 65 (10 self) - Add to MetaCart
Abstract not found
Powered by: Apache Solr
  • About CiteSeerX
  • Submit and Index Documents
  • Privacy Policy
  • Help
  • Data
  • Source
  • Contact Us

Developed at and hosted by The College of Information Sciences and Technology

© 2007-2019 The Pennsylvania State University