Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition

A new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented. The eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis. An efficient method for estimating the coefficients with respect to this basis is proposed. The method uses a Galerkin-based approach by orthogonalizing the residual in the eigenvalue–eigenvector equation to the subspace spanned by the basis functions used for approximation. In this way, the stochastic problem is framed as a system of deterministic non-linear algebraic equations. This system of equations is solved using a Newton–Raphson algorithm. Although the proposed approach is not based on statistical sampling, the efficiency of the proposed method can be significantly enhanced by initializing the non-linear iterative process with a small statistical sample synthesized through a Monte Carlo sampling scheme. The proposed method offers a number of advantages over existing methods based on statistical sampling. First, it provides an approximation to the complete probabilistic description of the eigensolution. Second, it reduces the computational overhead associated with solving the statistical eigenvalue problem. Finally, it circumvents the dependence of the statistical solution on the quality of the underlying random number generator. Copyright © 2007 John Wiley & Sons, Ltd.