Iterative solution of the random eigenvalue problem

Algebraic eigenvalue problems play an important role in a variety of fields. In structural mechanics, eigenvalue problems commonly appear in the context of, e.g., vibrations and buckling. In the case that the considered structure as well as the considered loading conditions are exactly known, i.e. deterministic, efficient and robust methods for the computation of eigenvalues and corresponding eigenvectors exist. In the more realistic case that the structure and the loading conditions are uncertain (described by random fields), eigenvalues and eigenvectors will also be uncertain. These random eigenvalues and random eigenvectors can then be determined by solving the random eigenvalue problem, for which the availability of efficient and robust methods is limited.

In this contribution an uncertainty analysis is performed for the random eigenvalue problem. The purpose of uncertainty analysis is to determine the statistical moments (mean, standard deviation, etc.) of the eigenvalues and eigenvectors. A projection on a spectral basis of Hermite polynomials is used to determine the statistical moments of the eigenvalues. The Galerkin method [

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] as used for linear algebraic problems turns out to yield an ill-posed system of equations. To avoid this problem, the deterministic inverse power method for the computation of eigenvalues is extended to yield the stochastic inverse power method.

The stochastic inverse power method consists of two steps. In the first step the eigenvector is updated using inverse iteration. In this step Galerkin’s method is used to determine the projection of the eigenvector on a spectral basis. In the second step the stochastic eigenvalue is updated using the Rayleighquotient. This sequence is repeated until both the stochastic eigenvalue and stochastic eigenvector have converged. The deterministic eigenvalue (with a small variation to avoid singularity problems) can be used as an initial setting for this iterative process.

The convergence of the iterative procedure can be controlled by partially updating the eigenvector. Numerical damping is then added to the system, preventing the solution from oscillating. In the case of moderate variations, the stochastic inverse power method turns out to be a robust method for the computation of the eigenvalues and eigenvectors of symmetric matrices. In the case of non-symmetric matrices, possible convergence problems can appear due to the fact that the spectral basis is not capable of spanning the (possibly non-differentiable) exact solution of the random eigenvalue problem.

The accuracy of the stochastic inverse power method (compared to aMonte-Carlo simulation) is demonstrated using numerical examples for the symmetric and non-symmetric problem.