Regularity and generalized polynomial chaos approximation of parametric and random second-order hyperbolic partial differential equations
Let \(D\) be a bounded Lipschitz domain in \(R^{d}\). In \(Q_{T}=(0,T)\times D\) the authors consider the stochastic wave equation \({\partial^2 u\over \partial t^2}-\nabla\cdot(a(x,\omega)\nabla u)=g(t,x)\), \(u|_{t=0}=g_1\), \(u_{t}|_{t=0}=g_2\). It is assumed that the coefficient \(a(x,\omega)\) is a random field on probability space \((\Omega,\Sigma,P)\) over \(L^{\infty}(D)\). The forcing \(g\) and initial data \(g_1\) and \(g_2\) are assumed to be deterministic. The authors show that the law of the random solution can be represented as a deterministic function of a countable number of coordinates. For a class of equations with regular right-hand side and compatible initial conditions it is shown that this solution is, as a function of the coordinates, smooth as mapping from the parameter domain into suitable Sobolev spaces in which deterministic wave equation is well-posed. The authors investigate the smoothness of the parametric solution in terms of Gervey regularity. Sufficient conditions for the \(p\)-summability of the generalized polynomial chaos expansion of the parametric solution in terms of the countably many input parameters are obtained and rates of convergence of best \(N\)-term polynomial chaos type approximations of the parametric solution are presented.
scientific article; zbMATH DE number 6073839