TY - GEN
DO - 10.5075/epfl-MATHICSE-263081
AB - 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.
T1 - MATHICSE Technical Report : A stochastic collocation method for the second order wave equation with a discontiuous random speed
DA - 2011-11-07
AU - Motamed, Mohammad
AU - Nobile, Fabio
AU - Tempone, Raúl
PB - MATHICSE
PP - Écublens
N1 - MATHICSE Technical Report Nr. 28.2011 November 2011
ID - 263081
KW - Stochastic partial differential equations
KW - Wave equation
KW - Collocation method
KW - Finite differences
KW - Finite elements
KW - Uncertainty quantification
KW - Error analysis
UR - http://infoscience.epfl.ch/record/263081/files/28.2011_MM-FN-RT.pdf
ER -