An Efficient Computational Solution Scheme of the Random Eigenvalue Problems

This paper presents a practical solution for probabilistic characterization of real valued eigenvalues of positive semi-definite random matrices. The present method is founded on the concept of high dimensional model representation (HDMR) technique. The method involves HDMR that facilitates lower dimensional approximation of the eigenvalues, response surface generation of HDMR component functions, and efficient Monte Carlo simulation for probability density functions. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Results of two numerical examples indicate that the proposed method provides accurate and computationally efficiency. Compared with commonly-used perturbation and recently-developed asymptotic methods, no derivatives of eigenvalues are required in the present method. Nomenclature x = N-dimensional random variables x = reference points for HDMR expansion xm = mean of random variable xs = standard deviation of random variable ()E · = expectation operator S x = covariance of random variables ()·k = stiffness matrix ()·m = mass matrix ()L · = eigenvalues ()F · = eigenvectors ()jf · = MLS interpolation function n = sample points in HDMR expansion s = order of component function in HDMR expansion l = order of HDMR expansion sN = simulation size in direct MCS I