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. 2012.
"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 = 2012,
month = 8
}

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 3

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