Recursive Gramian and Hankel map approximation of large dynamical systems.

In the last twenty years, model reduction of large scale dynamical systems has become very popular. The idea is to construct a “simple” lower order model that approximates well the behavior of the “complex” larger dynamical model. A complex system is essentially a mathematical model which describes a real world physical process. This mathematical model is often characterized by partial differential equations (PDEs). Since improved accuracy (using e.g. a very fine discretization) leads to large and sparse models of high complexity (see e.g. [11]), this may become prohibitive for certain computations (control, optimization, . . . ). Therefore it is essential to design models of reduced complexity. Most ideas developed for linear systems are based on the dominant spaces of Gramians [10] (energy functions for inand outgoing signals), which are the solutions of Lyapunov or Stein equations. A lot of work is still needed to efficiently compute these solutions (or their dominant spaces) when the system matrices are large and