Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion

The computational efficiency of random field representations with the Karhunen–Loève (KL) expansion relies on the solution of a Fredholm integral eigenvalue problem. This contribution compares different methods that solve this problem. Focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions. These include the Nyström method as well as collocation and Galerkin projection methods. Among the Galerkin methods, we investigate the finite element method (FEM) and propose the application of the finite cell method (FCM). This method is based on an extension to the FEM but avoids mesh generation on domains of complex geometric shape. The FCM was originally presented in Parvizian et al. (2007) [17] for the solution of elliptic boundary value problems. As an alternative to the L2L2-projection of the covariance function used in the Galerkin method, H1/2H1/2-projection and discrete projection are investigated. It is shown that the expansion optimal linear estimation (EOLE) method proposed in Li and Der Kiureghian (1993) [18] constitutes a special case of the Nyström method. It is found that the EOLE method is most efficient for the numerical solution of the KL expansion. The FEM and the FCM are more efficient than the EOLE method in evaluating a realization of the random field and, therefore, are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime – e.g., in finite element reliability analysis.