On accuracy of hierarchical eigenvalue/eigenvector approximations

We consider issues when designing a posteriori eigenvalue/eigenvector bounds for multi-scale problems in mechanics. Issues of cluster stability of eigenvalue estimate will receive a special treatment. This is a prerequisite for any comprehensive analysis of the subspace approximation problem. For a sake of being definite assume that we have constructed a vector $\psi$ which approximates the eigenvector $v$, $\|v\|=1$, $Hv=\lambda v$. Then, typically, one is satisfied with $\psi$ such that $\sin\angle(\psi,v)$ be "tiny". One expects that in such a situation $\psi$ can be used to extract spectral information on $H$. Unfortunately, small $\sin\angle(\psi,v)$ does not always imply that quality information on $\lambda$ can be extracted from $\psi$. Surely an approximate eigenvector which yields insufficient information on the accompanying eigenvalue cannot be considered as a satisfactory solution to a spectral approximation problem. The so-called "standard" or "relative" $\sin\Theta$ theory (Davis--Kahan, Ipsen, Li, Mathias--Veselić...) gives only a partial answer to this problem. Indeed, in a "singular" situation the relative residual is typically large (as it should be for a poor approximation $\psi$) but the estimated quantity $\sin\Theta(\psi,v)$ can be (and in the presented example it will be) very tiny. So we are in a situation in which we have to rely on a possibly very un-sharp inequality to obtain a correct hint on the "accuracy" of $\psi$. We reconsider the approximation problem in the geometry of the "energy" norm $\|\cdot\|_H$. The obtained estimates are an energy norm variant of the Mathias--Veselić eigenvector inequalities. We also present a new class of quadratic cluster robust estimates for eigenvalues. These estimates generalise the work of Drmač and Hari on a relative version of quadratic residual estimates. Furthermore, we show that all of the results hold for positive definite operators of Mathematical Physics both in bounded and unbounded domains. To evaluate the norm of the scaled residual we work in the dual of the "energy" space. Several techniques, which are based on the Green Function (e.g. Feynman--Katz formula, $\mathcal{H}$-inversion ...), will be explored in relation to this a posteriori estimation problem. As an illustration of the theory we will numerically tackle a couple of laboratory examples and compare results with other recent a posteriori eigenvalue estimates from the literature. This is a joint work with Zlatko Drmač and Krešimir Veselić.