Polynomial chaos enrichment for analyzing stochastic systems with clustered eigenvalues

The polynomial chaos decomposition has proven to be an efficient tool for characterizing the eigenspaces of stochastic dynamical with well-separated eigenvalues. For systems with closely spaced modes, difficulties arise due to the extreme sensitivity of the eigenvalues and corresponding eigenvectors to perturbations in the system dynamics, thus greatly taxing any polynomial approximation. An enrichment scheme to the standard polynomial chaos decomposition is proposed in this paper that permits the accurate resolution of this difficulty. Enrichment is provided through judiciously chosen functional forms with the enriched expansion providing a better stochastic characterization of the dominant eigenspace.