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

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

Journal Article · Tue May 19 00:00:00 EDT 2015 · Probabilistic Engineering Mechanics

DOI:https://doi.org/10.1016/j.probengmech.2015.05.002· OSTI ID:1235335

Emery, John M. ^[1]; Grigoriu, Mircea Dan ^[2]

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

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.

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: ASC

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1235335

Report Number(s):: SAND--2015-20740J; 558187

Journal Information:: Probabilistic Engineering Mechanics, Journal Name: Probabilistic Engineering Mechanics Journal Issue: C Vol. 41; ISSN 0266-8920

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (13)

Reduced order models for random functions. Application to stochastic problems Grigoriu, M. Applied Mathematical Modelling, Vol. 33, Issue 1 https://doi.org/10.1016/j.apm.2007.10.023	journal	January 2009
Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations Matthies, Hermann G.; Keese, Andreas Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 12-16 https://doi.org/10.1016/j.cma.2004.05.027	journal	April 2005
Multi-resolution analysis of Wiener-type uncertainty propagation schemes Le Maı̂tre, O. P.; Najm, H. N.; Ghanem, R. G. Journal of Computational Physics, Vol. 197, Issue 2 https://doi.org/10.1016/j.jcp.2003.12.020	journal	July 2004
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations Wan, Xiaoliang; Karniadakis, George Em Journal of Computational Physics, Vol. 209, Issue 2 https://doi.org/10.1016/j.jcp.2005.03.023	journal	November 2005
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications Foo, Jasmine; Wan, Xiaoliang; Karniadakis, George Em Journal of Computational Physics, Vol. 227, Issue 22 https://doi.org/10.1016/j.jcp.2008.07.009	journal	November 2008
A method for solving stochastic equations by reduced order models and local approximations Grigoriu, M. Journal of Computational Physics, Vol. 231, Issue 19 https://doi.org/10.1016/j.jcp.2012.06.013	journal	August 2012
On the accuracy of the polynomial chaos approximation Field, R. V.; Grigoriu, M. Probabilistic Engineering Mechanics, Vol. 19, Issue 1-2 https://doi.org/10.1016/j.probengmech.2003.11.017	journal	January 2004
Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients Todor, Radu Alexandru; Schwab, Christoph IMA Journal of Numerical Analysis, Vol. 27, Issue 2 https://doi.org/10.1093/imanum/drl025	journal	April 2007
Some Integrals Involving Hermite Polynomials Busbridge, Ida W. Journal of the London Mathematical Society, Vol. s1-23, Issue 2 https://doi.org/10.1112/jlms/s1-23.2.135	journal	April 1948
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Journal on Numerical Analysis, Vol. 45, Issue 3 https://doi.org/10.1137/050645142	journal	January 2007
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/060663660	journal	January 2008
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/070680540	journal	January 2008
Barycentric Lagrange Interpolation Berrut, Jean-Paul; Trefethen, Lloyd N. SIAM Review, Vol. 46, Issue 3 https://doi.org/10.1137/S0036144502417715	journal	January 2004

Cited By (1)

Tsunami hazard assessments with consideration of uncertain earthquake slip distribution and location: TSUNAMI HAZARD AND UNCERTAIN EARTHQUAKES Sepúlveda, Ignacio; Liu, Philip L. -F.; Grigoriu, Mircea Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 9 https://doi.org/10.1002/2017jb014430	journal	September 2017

Similar Records

A method for solving stochastic equations by reduced order models and local approximations

Journal Article · Wed Aug 01 00:00:00 EDT 2012 · Journal of Computational Physics · OSTI ID:22192332

The Galerkin-collocation method for hyperbolic initial boundary value problems

Journal Article · Wed Jun 01 00:00:00 EDT 1994 · Journal of Computational Physics; (United States) · OSTI ID:7163969

Sparse grid collocation schemes for stochastic natural convection problems

Journal Article · Sun Jul 01 00:00:00 EDT 2007 · Journal of Computational Physics · OSTI ID:20991599

Related Subjects

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

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

Citation Formats

References (13)

Cited By (1)

Similar Records

Related Subjects