Computation of Smallest Eigenvalues using Spectral Schur Complements

The automated multilevel substructuring method (AMLS) was recently presented as an alternative to well-established methods for computing eigenvalues of large matrices in the context of structural engineering. This technique is based on exploiting a high level of dimensional reduction via domain decomposition and projection methods. This paper takes a purely algebraic look at the method and explains that it can be viewed as a combination of three ingredients: (a) A first order expansion to a nonlinear eigenvalue problem that approximates the restriction of the original eigenproblem on the interface between the subdomains, (b) judicious projections on partial eigenbases that correspond to the interior of the subdomains, (c) recursivity. This viewpoint leads us to explore variants of the method which use Krylov subspaces instead of eigenbases to construct subspaces of approximants. The nonlinear eigenvalue problem viewpoint yields a second order approximation as an enhancement to the first order technique inherent to AMLS. Numerical experiments are reported to validate the approaches presented.