New formulation of Cholesky decomposition and applications in stochastic simulation

Abstract

Monte Carlo simulation plays a significant role in the mechanical and structural analysis due to its versatility and accuracy. Classical spectral representation method is based on the direct decomposition of the power spectral density (PSD) or evolutionary power spectral density (EPSD) matrix through Cholesky decomposition. This direct decomposition of complex matrix usually results in large computational time and storage memory.In this study, a new formulation of the Cholesky decomposition for the EPSD/PSD matrix and corresponding simulation scheme are presented. The key idea to this approach is to separate the phase from the complex EPSD/PSD matrix. The derived real modulus matrix evidently expedites decomposition compared to the direct Cholesky decomposition of the complex EPSD/PSD matrix. In the proposed simulation scheme, the separated phase can be easily assembled. The modulus of EPSD/PSD matrix could be further decomposed into the modulus of coherence matrix (or lagged coherence matrix), which describes the basic coherence structure of stochastic process. The lagged coherence matrix is independence of time and thus remarkably improves the Cholesky decomposition efficiency.The application of the proposed schemes to Gaussian stochastic simulations is presented. Firstly, the previous closed-form wind speed simulation algorithm for equally-spaced locations is extended to a more general situation. Secondly, the proposed approach facilitates the application of interpolation technique in stochastic simulation. The application of interpolation techniques in the wind field simulation is studied as an example.