Abstract

We analyze the stochastic initial-boundary value problem for the elastic
wave equation with random coefficients and deterministic data. We
propose a stochastic collocation method for computing statistical
moments of the solution or statistics of some given quantities of
interest. We study the convergence rate of the error in the stochastic
collocation method. In particular, we show that, the rate of convergence
depends on the regularity of the solution or the quantity of interest
in the stochastic space, which is in turn related to the regularity of
the deterministic

data in the physical space and the type of the
quantity of interest. We demonstrate that a fast rate of convergence is
possible in two cases: for the elastic wave solutions with high regular
data; and for some high regular quantities of interest even in the
presence of low regular data. We perform numerical examples, including a
simplified earthquake, which confirm the analysis and show that the
collocation method is a valid alternative to the more traditionalMonte
Carlo sampling method for problems with high stochastic regularity.