Simulation of multivariate stationary Gaussian stochastic processes: Hybrid spectral representation and proper orthogonal decomposition approach

By observing that the optimal basis for the proper orthogonal decomposition (POD) can be obtained from the cross power spectral density (XPSD) matrix of a multivariate stationary Gaussian stochastic process, the computational efficiency, in both time and memory consumption, of simulations of this process is improved by using a hybrid spectral representation and POD approach with negligible loss of accuracy. This hybrid approach actually simulates another multivariate process with many fewer variables in an optimal subspace obtained by the POD. This approach is straightforward, effective, and does not place any conditions on the XPSD matrices. Furthermore, the error induced by the reduction of variables is predictable and controllable prior to the simulation procedure. The spectral representation method (SRM) is discussed in a heuristic way. In this paper, a specific POD theorem is formally stated, proved, and related to XPSD matrices. A numerical example is given to demonstrate the effectiveness of this hybrid approach. This approach may also have potential applications for simulations of nonstationary non-Gaussian processes.