A Practical Solution of the Random Eigenvalue Problems using Factorized Decomposition Technique

This paper presents a practical solution for probabilistic characterization of real valued eigenvalues of positive semi-definite random matrices. The method involves a novel dimension reduction technique that facilitates a lower-dimensional approximation of a high dimensional problem. The present method is basically founded on the idea of high dimensional model representation (HDMR) technique. HDMR is a multivariate representation to capture the input-output relationship of a physical system with many 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. Practically for most well-defined physical systems, only relatively low order correlations of the input variables are expected to have a significant effect on the overall response. HDMR expansion utilizes this property to present an accurate hierarchical representation of the physical system. The method involves multiplicative decomposition of a multivariate eigenfunction into multiple one-dimensional eigenvalues. The effort required by the proposed method can be viewed as performing few deterministic analyses at selected input defined by sample points. Compared with commonly used perturbation and recently developed asymptotic methods, no derivatives of eigenvalues are required by the new method developed. Results of numerical examples from structural dynamics indicate that the decomposition method provides sufficient accuracy for estimation of probability densities of eigenvalues for various cases including closely-spaced modes and large statistical variations of input.