# Operator compression preserving lowest energy eigenspace.

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.

• You do not need to form the matrix to run Lanczos, you just need a "black-box" function that computes $Av$ given a vector $v$. If you cannot even do this efficiently, please specify how you can interact with the matrix, or which special structure it has. May 6 '11 at 7:03
• Yes, you only need to form $Av$ to do Lanczos. My matrix is exactly this $\mathbb{H}$: en.wikipedia.org/wiki/Configuration_interaction . I would like it to be smaller while still preserving the lowest energy states. A similar problem arises in finite element simulations. Consider coarsening the discretization in certain regions in order to work with a smaller stiffness matrix. Normally this is done using geometric intuition but in my problem the space is high dimensional so I would like to somehow automate the coarsening. May 6 '11 at 21:35