| Model reduction of the quadratic eigenvalue problem is an area of considerable recent interest. Such eigenvalue problems arise in control theory applications, acoustics, and structural analysis, where the dominant behavior of the system is determined by a relatively small number of eigenvalues. As such models can be very large, directly computing all eigenpairs is impractical. Instead, the system is reduced to a matrix equation of much smaller degree which is computationally amenable. Customarily, this is done through a linearization procedure.In this thesis, we present model reduction techniques that construct a reduced-order model which is also given by a quadratic eigenvalue problem. In chapter 2, we describe a Krylov-type projection method that reduces a symmetric monic quadratic eigenvalue problem to another symmetric QEP of banded structure. We also describe a Rayleigh-Ritz procedure for subspace enlargement which accelerates convergence of the projected problem to a desired eigenpair. In chapter 3, we examine several linearization techniques and their expected rates of convergence using a moment-matching result of Grimme. In addition, we develop a variant of nonsymmetric Lanczos that reduces a monic QEP to one of triangular-Hessenberg form, with optimal orders of moment-matching. |
