The Unifying Feature of Projection in Model Order Reduction

This paper considers the problem of model order reduction of linear systems with the emphasis on the common features of the main approaches. One of these features is the unifying role of operator projection in model reduction. It is shown how projections are implemented for different methods of model reduction and what their properties are. The other common feature is the subspaces where projections are defined. The main approaches for model reduction which are considered in the paper are balanced truncation, proper orthogonal decomposition and the Lanczos procedure from the Krylov subspace methods. It is shown that the range spaces of system gramians for balanced truncation and the range space of the reachability and observability matrices for the Lanczos procedure coincide. The connection between balanced truncation and the proper orthogonal decomposition method is also established. Therefore, the methods for model reduction are similar in terms of general operational principles, and differ mostly in their technical implementation. Several numerical examples are considered showing the validity of the proposed conjectures.

NUCLEAR PHYSICS WITH STATISTICS

Properties of shell model Hamiltonians, and in particular the tri-diagonal matrices obtained with the Lanczos procedure are exploited to determine the density of states and to estimate the ground-state energies. It will be shown that with a few parameters that are adjusted to the moments up to fourth order of the Hamiltonian, one can model the resultant tri-diagonal matrices. An estimate of the ground state energy can quickly be obtained from the modeled matrices, and the density of states for the full matrix can quickly be obtained using the WKB approximation. The accuracy of the ground state is dependent on high moments, six and greater.