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000263081 001__ 263081 000263081 005__ 20190619220150.0 000263081 0247_ $$2doi$$a10.5075/epfl-MATHICSE-263081 000263081 037__ $$aREP_WORK 000263081 245__ $$aMATHICSE Technical Report : A stochastic collocation method for the second order wave equation with a discontiuous random speed 000263081 269__ $$a2011-11-07 000263081 260__ $$aÉcublens$$c2011-11-07$$bMATHICSE 000263081 300__ $$a48 000263081 336__ $$aWorking Papers 000263081 500__ $$aMATHICSE Technical Report Nr. 28.2011 November 2011 000263081 520__ $$aIn 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. 000263081 6531_ $$aStochastic partial differential equations 000263081 6531_ $$aWave equation 000263081 6531_ $$aCollocation method 000263081 6531_ $$aFinite differences 000263081 6531_ $$aFinite elements 000263081 6531_ $$aUncertainty quantification 000263081 6531_ $$aError analysis 000263081 700__ $$aMotamed, Mohammad 000263081 700__ $$g118353$$aNobile, Fabio$$0241873 000263081 700__ $$aTempone, Raúl 000263081 710__ $$aMATHICSE-Group 000263081 787__ $$whttps://infoscience.epfl.ch/record/180196$$eIs Previous Version Of 000263081 8560_ $$ffabio.nobile@epfl.ch 000263081 8564_ $$yVersion 1$$zVersion 1$$uhttps://infoscience.epfl.ch/record/263081/files/28.2011_MM-FN-RT.pdf$$s1629079 000263081 909C0 $$zJunod, Julien$$xU12495$$pCSQI$$mrachel.bordelais@epfl.ch$$0252411 000263081 909CO $$qGLOBAL_SET$$pworking$$pDOI$$pSB$$ooai:infoscience.epfl.ch:263081 000263081 960__ $$afabio.nobile@epfl.ch 000263081 961__ $$apierre.devaud@epfl.ch 000263081 973__ $$aEPFL 000263081 980__ $$aREP_WORK 000263081 981__ $$aoverwrite