The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition formula is proven. This formula 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 are derived in the trace class and Hilbert--Schmidt norms with numerical simulations illustrating the results.

翻译：考虑了对线性随机部分差分方程的微量溶解的共差操作员的近似共差问题。将得出一个包含微量溶解的半组的整体方程式,并证明有一个一般的误差分解公式。该公式适用于悬浮对立分解方程式和随机波方程的共差操作员的近似差,两者均在封闭的域内。近似值基于空间的有限元素分解和时速指数函数的合理近似。聚合率来自跟踪类和Hilbert-Schmidt规范,并用数字模拟来说明结果。