Analysis and computation of the elastic wave equation with random coefficients

Mohammad Motamed; Fabio Nobile; Raul Tempone

doi:10.1016/j.camwa.2015.09.013

Analysis and computation of the elastic wave equation with random coefficients

Mohammad Motamed, Fabio Nobile, Raul Tempone

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21 Scopus citations

Abstract

We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

Original language	English (US)
Pages (from-to)	2454-2473
Number of pages	20
Journal	Computers & Mathematics with Applications
Volume	70
Issue number	10
DOIs	https://doi.org/10.1016/j.camwa.2015.09.013
State	Published - Oct 21 2015

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 "Bayesian earthquake source validation for ground motion simulation" is also acknowledged. The third author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. The second author has been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".

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title = "Analysis and computation of the elastic wave equation with random coefficients",

abstract = "We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.",

author = "Mohammad Motamed and Fabio Nobile 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 {"}Bayesian earthquake source validation for ground motion simulation{"} is also acknowledged. The third author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. The second author has been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) {"}Advanced numerical techniques for uncertainty quantification in engineering and life science problems{"}.",

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doi = "10.1016/j.camwa.2015.09.013",

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AU - Motamed, Mohammad

AU - Nobile, Fabio

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 "Bayesian earthquake source validation for ground motion simulation" is also acknowledged. The third author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. The second author has been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".

PY - 2015/10/21

Y1 - 2015/10/21

N2 - We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

AB - We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for approximating quantities with high stochastic regularity.

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JO - Computers & Mathematics with Applications

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ER -

Analysis and computation of the elastic wave equation with random coefficients

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