A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations.
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| Title: | A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations. |
|---|---|
| Authors: | Vidal-Codina, F.1 fvidal@mit.edu, Nguyen, N.C.1 cuongng@mit.edu, Giles, M.B.2 mike.giles@maths.ox.ac.uk, Peraire, J.1 peraire@mit.edu |
| Source: | Journal of Computational Physics. Sep2015, Vol. 297, p700-720. 21p. |
| Subjects: | Partial differential equations, Stochastic analysis, Discretization methods, Parameterization, Stochastic convergence, Monte Carlo method |
| Abstract: | We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 103654539 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Vidal-Codina%2C+F%2E%22">Vidal-Codina, F.</searchLink><relatesTo>1</relatesTo><i> fvidal@mit.edu</i><br /><searchLink fieldCode="AR" term="%22Nguyen%2C+N%2EC%2E%22">Nguyen, N.C.</searchLink><relatesTo>1</relatesTo><i> cuongng@mit.edu</i><br /><searchLink fieldCode="AR" term="%22Giles%2C+M%2EB%2E%22">Giles, M.B.</searchLink><relatesTo>2</relatesTo><i> mike.giles@maths.ox.ac.uk</i><br /><searchLink fieldCode="AR" term="%22Peraire%2C+J%2E%22">Peraire, J.</searchLink><relatesTo>1</relatesTo><i> peraire@mit.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Computational+Physics%22">Journal of Computational Physics</searchLink>. Sep2015, Vol. 297, p700-720. 21p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+analysis%22">Stochastic analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Parameterization%22">Parameterization</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+convergence%22">Stochastic convergence</searchLink><br /><searchLink fieldCode="DE" term="%22Monte+Carlo+method%22">Monte Carlo method</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Computational Physics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jcp.2015.05.041 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 21 StartPage: 700 Subjects: – SubjectFull: Partial differential equations Type: general – SubjectFull: Stochastic analysis Type: general – SubjectFull: Discretization methods Type: general – SubjectFull: Parameterization Type: general – SubjectFull: Stochastic convergence Type: general – SubjectFull: Monte Carlo method Type: general Titles: – TitleFull: A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Vidal-Codina, F. – PersonEntity: Name: NameFull: Nguyen, N.C. – PersonEntity: Name: NameFull: Giles, M.B. – PersonEntity: Name: NameFull: Peraire, J. IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 09 Text: Sep2015 Type: published Y: 2015 Identifiers: – Type: issn-print Value: 00219991 Numbering: – Type: volume Value: 297 Titles: – TitleFull: Journal of Computational Physics Type: main |
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