Application of Proper Orthogonal Decomposition in Fast Fourier Transform—Assisted Multivariate Nonstationary Process Simulation

Abstract

The classic spectral representation method (SRM)-based nonstationary process simulation algorithm is used extensively in the engineering community. However, it is less efficient owing to the unavailability of fast Fourier transform (FFT). In this paper, an efficient, almost accurate, and straightforward algorithm is developed for the simulation of the multivariate nonstationary process. In this method, an evolutionary spectral matrix is decomposed via Cholesky method, and then proper orthogonal decomposition (POD) is used to factorize decomposed spectra as the summation of the products of time and frequency functions. Because original time-dependent decomposed spectra are decoupled via factorization, FFT can be used to significantly expedite the simulation efficiency. This POD-based factorization is totally data-driven and optimal, and fewer items are required in matching decomposed spectra. Therefore, the accuracy and efficiency of the factorization can be guaranteed at the same time. Another attractive feature of this factorization is straightforwardness, because only regular eigenvector decomposition is involved. Numerical examples of nonstationary processes are used to demonstrate the effectiveness and accuracy of the proposed approach. Results show that the factorization and simulation agree with the targets very well. In addition, the speed at which sample functions are generated is significantly improved over classic SRM, in which the full summation of sine/cosine terms is needed.