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    A stochastic collocation method for the second order wave equation with a discontinuous random speed

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    Type
    Article
    Authors
    Motamed, Mohammad
    Nobile, Fabio
    Tempone, Raul cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Stochastic Numerics Research Group
    Date
    2012-08-31
    Online Publication Date
    2012-08-31
    Print Publication Date
    2013-03
    Permanent link to this record
    http://hdl.handle.net/10754/562285
    
    Metadata
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    Abstract
    In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. © 2012 Springer-Verlag.
    Citation
    Motamed, M., Nobile, F., & Tempone, R. (2012). A stochastic collocation method for the second order wave equation with a discontinuous random speed. Numerische Mathematik, 123(3), 493–536. doi:10.1007/s00211-012-0493-5
    Sponsors
    This work was supported by the King Abdullah University of Science and Technology (AEA project "Bayesian earthquake source validation for ground motion simulation"), the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar", and the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). The second author was partially supported by the Italian grant FIRB-IDEAS (Project no. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".
    Publisher
    Springer Nature
    Journal
    Numerische Mathematik
    DOI
    10.1007/s00211-012-0493-5
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00211-012-0493-5
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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