Simplified CSP analysis of a stiff stochastic ODE system

Maher Salloum; Alen Alexanderian; Olivier P. Le Maître; Habib N. Najm; Omar M. Knio

doi:10.1016/j.cma.2012.01.001

Simplified CSP analysis of a stiff stochastic ODE system

Maher Salloum, Alen Alexanderian, Olivier P. Le Maître, Habib N. Najm, Omar M. Knio^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

Original language	English (US)
Pages (from-to)	121-138
Number of pages	18
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	217-220
DOIs	https://doi.org/10.1016/j.cma.2012.01.001
State	Published - Apr 1 2012
Externally published	Yes

Bibliographical note

Funding Information:
This work was supported by the US Department of Energy (DOE) under Award No. DE-SC0001980. HNN was supported by the DOE Office of Basic Energy Sciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences . Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000. This report was prepared as an account of work sponsored in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The work of OLM. is partially supported by the French Agence Nationale pour la Recherche (Project ANR-2010-Blan-0904) and the GNR MoMaS funded by ANDRA , BRGM , CEA , EDF , and IRSN .

Keywords

CSP
Polynomial chaos
Random eigenvalues
Stiff system
Uncertain ODE

ASJC Scopus subject areas

Computational Mechanics
Mechanics of Materials
Mechanical Engineering
General Physics and Astronomy
Computer Science Applications

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10.1016/j.cma.2012.01.001

Cite this

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title = "Simplified CSP analysis of a stiff stochastic ODE system",

abstract = "We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.",

keywords = "CSP, Polynomial chaos, Random eigenvalues, Stiff system, Uncertain ODE",

author = "Maher Salloum and Alen Alexanderian and {Le Ma{\^i}tre}, {Olivier P.} and Najm, {Habib N.} and Knio, {Omar M.}",

note = "Funding Information: This work was supported by the US Department of Energy (DOE) under Award No. DE-SC0001980. HNN was supported by the DOE Office of Basic Energy Sciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences . Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000. This report was prepared as an account of work sponsored in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The work of OLM. is partially supported by the French Agence Nationale pour la Recherche (Project ANR-2010-Blan-0904) and the GNR MoMaS funded by ANDRA , BRGM , CEA , EDF , and IRSN . ",

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AU - Alexanderian, Alen

AU - Le Maître, Olivier P.

AU - Najm, Habib N.

AU - Knio, Omar M.

N1 - Funding Information: This work was supported by the US Department of Energy (DOE) under Award No. DE-SC0001980. HNN was supported by the DOE Office of Basic Energy Sciences (BES) Division of Chemical Sciences, Geosciences, and Biosciences . Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94-AL85000. This report was prepared as an account of work sponsored in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The work of OLM. is partially supported by the French Agence Nationale pour la Recherche (Project ANR-2010-Blan-0904) and the GNR MoMaS funded by ANDRA , BRGM , CEA , EDF , and IRSN .

PY - 2012/4/1

Y1 - 2012/4/1

N2 - We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

AB - We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

KW - CSP

KW - Polynomial chaos

KW - Random eigenvalues

KW - Stiff system

KW - Uncertain ODE

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DO - 10.1016/j.cma.2012.01.001

M3 - Article

AN - SCOPUS:84856896956

SN - 0045-7825

VL - 217-220

SP - 121

EP - 138

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -

Simplified CSP analysis of a stiff stochastic ODE system

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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