263081
20190619220150.0
doi
10.5075/epfl-MATHICSE-263081
REP_WORK
MATHICSE Technical Report : A stochastic collocation method for the second order wave equation with a discontiuous random speed
2011-11-07
Écublens
MATHICSE
2011-11-07
48
Working Papers
MATHICSE Technical Report Nr. 28.2011 November 2011
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.
Stochastic partial differential equations
Wave equation
Collocation method
Finite differences
Finite elements
Uncertainty quantification
Error analysis
Motamed, Mohammad
241873
Nobile, Fabio
118353
Tempone, Raúl
MATHICSE-Group
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https://infoscience.epfl.ch/record/180196
fabio.nobile@epfl.ch
1629079
http://infoscience.epfl.ch/record/263081/files/28.2011_MM-FN-RT.pdf
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Version 1
252411
rachel.bordelais@epfl.ch
CSQI
U12495
Junod, Julien
oai:infoscience.epfl.ch:263081
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fabio.nobile@epfl.ch
pierre.devaud@epfl.ch
EPFL
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