We consider the problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation (SPDE). The SPDE is driven by space-time Gaussian noise in the form of a cylindrical Wiener process, a Q-Wiener process being a special case thereof.
An integral equation involving the semigroup of the mild solution is derived along with a general error decomposition formula. This is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates in the trace class and Hilbert–Schmidt norms are derived, and we compare these to the strong and weak convergence rates of the corresponding approximations of the solution to the SPDE.
We also describe some new results related to how the regularity of a kernel of the driving Q-Wiener process relates to regularity of Q as measured in terms of fractional powers of the generator of the semigroup. We discuss how this affects the convergence rates of the considered approximations. Important examples that fit into our framework include the class of Matérn kernels. Numerical simulations illustrate the results.
This presentation is based on joint work with Mihály Kovács and Annika Lang.