Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Joakim Beck; Fabio Nobile; Lorenzo Tamellini; Raul Tempone

doi:10.1016/j.camwa.2013.03.004

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Joakim Beck, Fabio Nobile, Lorenzo Tamellini, Raul Tempone

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

50 Scopus citations

Abstract

In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved.

Original language	English (US)
Pages (from-to)	732-751
Number of pages	20
Journal	Computers & Mathematics with Applications
Volume	67
Issue number	4
DOIs	https://doi.org/10.1016/j.camwa.2013.03.004
State	Published - Mar 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems". The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

ASJC Scopus subject areas

Modeling and Simulation
Computational Theory and Mathematics
Computational Mathematics

Access to Document

10.1016/j.camwa.2013.03.004

Cite this

@article{de532039b51a452b9398f8839db2a0a9,

title = "Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients",

abstract = "In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. {\textcopyright} 2013 Elsevier Ltd. All rights reserved.",

author = "Joakim Beck and Fabio Nobile and Lorenzo Tamellini and Raul Tempone",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project {"}Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar{"} and King Abdullah University of Science and Technology (KAUST) AEA project {"}Predictability and Uncertainty Quantification for Models of Porous Media{"} is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) {"}Advanced numerical techniques for uncertainty quantification in engineering and life science problems{"}. The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.",

year = "2014",

month = mar,

doi = "10.1016/j.camwa.2013.03.004",

language = "English (US)",

volume = "67",

pages = "732--751",

journal = "Computers & Mathematics with Applications",

issn = "0898-1221",

publisher = "Elsevier",

number = "4",

}

TY - JOUR

T1 - Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

AU - Beck, Joakim

AU - Nobile, Fabio

AU - Tamellini, Lorenzo

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems". The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

PY - 2014/3

Y1 - 2014/3

N2 - In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved.

AB - In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane CN. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. © 2013 Elsevier Ltd. All rights reserved.

UR - http://hdl.handle.net/10754/563416

UR - https://linkinghub.elsevier.com/retrieve/pii/S0898122113001569

UR - http://www.scopus.com/inward/record.url?scp=84893838003&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2013.03.004

DO - 10.1016/j.camwa.2013.03.004

M3 - Article

SN - 0898-1221

VL - 67

SP - 732

EP - 751

JO - Computers & Mathematics with Applications

JF - Computers & Mathematics with Applications

IS - 4

ER -

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this