A stochastic collocation method for the second order wave equation with a discontinuous random speed
KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Stochastic Numerics Research Group
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AbstractIn 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.
SponsorsThis 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".