A method for solving stochastic equations by reduced order models and local approximations
Abstract
A method is proposed for solving equations with random entries, referred to as stochastic equations (SEs). The method is based on two recent developments. The first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments. The second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector, referred to as stochastic reduced order model (SROM), and uses it to construct a SROM for the solution of this equation. The proposed method is a direct extension of these two methods. It uses SROMs to select expansion points, rather than selecting these points by geometrical considerations, and represents the solution by linear and/or higher order local approximations. The implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic/differential equations. The method is conceptually simple, nonintrusive, efficient relative to classical Monte Carlo simulation, accurate, and guaranteed to converge to the exact solution.
 Authors:
 Cornell University, Ithaca, NY 148533501 (United States)
 Publication Date:
 OSTI Identifier:
 22192332
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 231; Journal Issue: 19; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPUTERIZED SIMULATION; DIFFERENTIAL EQUATIONS; EIGENVALUES; EXACT SOLUTIONS; MATHEMATICAL MODELS; MONTE CARLO METHOD; PERFORMANCE; PROBABILITY; RANDOMNESS; STOCHASTIC PROCESSES
Citation Formats
Grigoriu, M., Email: mdg12@cornell.edu. A method for solving stochastic equations by reduced order models and local approximations. United States: N. p., 2012.
Web. doi:10.1016/J.JCP.2012.06.013.
Grigoriu, M., Email: mdg12@cornell.edu. A method for solving stochastic equations by reduced order models and local approximations. United States. doi:10.1016/J.JCP.2012.06.013.
Grigoriu, M., Email: mdg12@cornell.edu. Wed .
"A method for solving stochastic equations by reduced order models and local approximations". United States.
doi:10.1016/J.JCP.2012.06.013.
@article{osti_22192332,
title = {A method for solving stochastic equations by reduced order models and local approximations},
author = {Grigoriu, M., Email: mdg12@cornell.edu},
abstractNote = {A method is proposed for solving equations with random entries, referred to as stochastic equations (SEs). The method is based on two recent developments. The first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments. The second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector, referred to as stochastic reduced order model (SROM), and uses it to construct a SROM for the solution of this equation. The proposed method is a direct extension of these two methods. It uses SROMs to select expansion points, rather than selecting these points by geometrical considerations, and represents the solution by linear and/or higher order local approximations. The implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic/differential equations. The method is conceptually simple, nonintrusive, efficient relative to classical Monte Carlo simulation, accurate, and guaranteed to converge to the exact solution.},
doi = {10.1016/J.JCP.2012.06.013},
journal = {Journal of Computational Physics},
number = 19,
volume = 231,
place = {United States},
year = {Wed Aug 01 00:00:00 EDT 2012},
month = {Wed Aug 01 00:00:00 EDT 2012}
}

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two wellestablished and successful approaches for solving general stochastic problems. A recently developed method based on stochastic reduced order models (SROMs) can also be used. Herein we provide a comparison of the three methods for some numerical examples; our evaluation only holds for the examples considered in the paper. The purpose of the comparisons is not to criticize the SC or SG methods, which have proven very useful for a broad range of applications, nor is it to provide overall ratings of these methods as compared to the SROM method.more »Cited by 4

A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions
In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix for generalized hat functions on the finite interval [0,T] is obtained. By using these basis functions and their stochastic operational matrix, such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution. Also, the rate of convergence of the proposed method is considered and it has been shown that it is O(1/(n{sup 2}) ). Further, in order to show themore » 
An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics
Because of the nonlinearity, closedform solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itôintegration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Errormore » 
FD method of arbitrary uniform order of accuracy for solving singularly perturbed boundary problems for secondorder ordinary differential equations
Using a functionaldiscrete approach, threepoint difference schemes of arbitrary order of accuracy are constructed for solving the Dirichlet problem for secondorder ordinary differential equations (ODE) with a small parameter multiplying the leading derivative. The uniform convergence of the schemes with respect to the small parameter is proved, and a recursive algorithm for their realization is constructed.