1 Skills, Concepts and Models in a Linear Algebra Course

By Ana Moura and Pedro A. Santos.

In recent years, there has been a lot of interest in order reduction of largescale linear dynamical systems. Many of the widely-used methods today employ some form of projection onto suitably chosen block Krylov subspaces. It is well understood that numerically robust techniques for constructing bases for these block Krylov subspaces need to be able to handle deflation of linearly dependent or nearly linearly dependent Krylov vectors; the case of linearly dependent vectors is called exact deflation, the case of nearly linearly dependent vectors is called inexact deflation. It is also well known that, at least in exact arithmetic, the occurrence of exact deflation is a desirable event, in the sense that it increases the accuracy of the reduced-order model. On the other hand, in order to have numerical stable procedures, in finiteprecision arithmetic, one needs to perform inexact deflation that in turn decreases the accuracy of the reduced-order model. In this talk, we discuss the effects of inexact deflation in projection-based order reduction. We review some of the underlying theoretical results about exact deflation, discuss some practical remedies to minimize the loss of accuracy in the case of inexact deflation, and present results of numerical experiments. We will also consider the special case of structure-preserving order reduction techniques, such as SPRIM, that employ suitably chosen partitionings of the underlying block Krylov subspaces in order to preserve key structures of the original large-scale linear dynamical system.