I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a way that $H'$ and $H$ have nearly identical eigenvectors and eigenvalues for the lowest 5 or 6 eigenvalues.

This could be done if I knew the lowest eigenvectors of $H$ as I could simply restrict $H$ to the space spanned by these values. However I would like to be able to find an approximation before solving eigenvalue problem (in fact my eventual goal is to make solving the eigenvalue problem more efficient).

I know of the recent work on approximating SDD systems with graph sparsification as well as multilevel operator compression. What else is out there?

The application is a full-CI treatment of a multiparticle quantum system.