On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Emery, John M.; Grigoriu, Mircea Dan

doi:10.1016/j.probengmech.2015.05.002

Title: On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Abstract

The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established 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. Furthermore, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.

Authors:

Richard V. Field, Jr.; Emery, John M. ^[1]; Grigoriu, Mircea Dan ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Cornell Univ., Ithaca, NY (United States)

Publication Date:: Tue May 19 00:00:00 EDT 2015

Research Org.:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Org.:: ASC

OSTI Identifier:: 1235335

Report Number(s):: SAND-2015-20740J
Journal ID: ISSN 0266-8920; 558187

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Accepted Manuscript

Journal Name:: Probabilistic Engineering Mechanics

Additional Journal Information:: Journal Volume: 41; Journal Issue: C; Journal ID: ISSN 0266-8920

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; approximation theory; Monte Carlo simulation; random variables and fields; stochastic differential equations; uncertainty propagation

Citation Formats


                    Richard V. Field, Jr., Emery, John M., and Grigoriu, Mircea Dan. On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems.  United States: N. p., 2015. 
Web.  doi:10.1016/j.probengmech.2015.05.002.

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                    Richard V. Field, Jr., Emery, John M., & Grigoriu, Mircea Dan. On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems.  United States.  https://doi.org/10.1016/j.probengmech.2015.05.002

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                    Richard V. Field, Jr., Emery, John M., and Grigoriu, Mircea Dan. Tue .  
"On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems".  United States.  https://doi.org/10.1016/j.probengmech.2015.05.002.  https://www.osti.gov/servlets/purl/1235335.

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@article{osti_1235335,

  title        = {On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems},

  author       = {Richard V. Field, Jr. and Emery, John M. and Grigoriu, Mircea Dan},

  abstractNote = {The stochastic collocation (SC) and stochastic Galerkin (SG) methods are two well-established 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. Furthermore, our objectives are to present the SROM method as an alternative approach to solving stochastic problems and provide information on the computational effort required by the implementation of each method, while simultaneously assessing their performance for a collection of specific problems.},

  doi          = {10.1016/j.probengmech.2015.05.002},

  journal      = {Probabilistic Engineering Mechanics},

  number       = C,

  volume       = 41,

  place        = {United States},

  year         = {Tue May 19 00:00:00 EDT 2015},

  month        = {Tue May 19 00:00:00 EDT 2015}

}

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https://doi.org/10.1016/j.probengmech.2015.05.002

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Works referenced in this record:

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
journal, January 2007

Babuška, Ivo; Nobile, Fabio; Tempone, Raúl
SIAM Journal on Numerical Analysis, Vol. 45, Issue 3
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A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
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Nobile, F.; Tempone, R.; Webster, C. G.
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Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations
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Matthies, Hermann G.; Keese, Andreas
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Reduced order models for random functions. Application to stochastic problems
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Grigoriu, M.
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A method for solving stochastic equations by reduced order models and local approximations
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Grigoriu, M.
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DOI: 10.1016/j.jcp.2012.06.013

An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
journal, January 2008

Nobile, F.; Tempone, R.; Webster, C. G.
SIAM Journal on Numerical Analysis, Vol. 46, Issue 5
DOI: 10.1137/070680540

The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
journal, November 2008

Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em
Journal of Computational Physics, Vol. 227, Issue 22
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Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
journal, April 2007

Todor, Radu Alexandru; Schwab, Christoph
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Multi-resolution analysis of Wiener-type uncertainty propagation schemes
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Le Maı̂tre, O. P.; Najm, H. N.; Ghanem, R. G.
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An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
journal, November 2005

Wan, Xiaoliang; Karniadakis, George Em
Journal of Computational Physics, Vol. 209, Issue 2
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Some Integrals Involving Hermite Polynomials
journal, April 1948

Busbridge, Ida W.
Journal of the London Mathematical Society, Vol. s1-23, Issue 2
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On the accuracy of the polynomial chaos approximation
journal, January 2004

Field, R. V.; Grigoriu, M.
Probabilistic Engineering Mechanics, Vol. 19, Issue 1-2
DOI: 10.1016/j.probengmech.2003.11.017

Works referencing / citing this record:

Tsunami hazard assessments with consideration of uncertain earthquake slip distribution and location: TSUNAMI HAZARD AND UNCERTAIN EARTHQUAKES
journal, September 2017

Sepúlveda, Ignacio; Liu, Philip L. -F.; Grigoriu, Mircea
Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 9
DOI: 10.1002/2017jb014430

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Title: On the efficacy of stochastic collocation, stochastic Galerkin, and stochastic reduced order models for solving stochastic problems

Abstract

Citation Formats

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