Cross-gramian based model reduction for datasparse systems
Younès Chahlaoui, Ulrike Baurt, Peter Benner · Research Explorer (The University of Manchester) · 2008
Model order reduction (MOR) is common in simulation, control and optimization of complex dynamical systems arising in modeling ofphysical processes, and in the spatial discretization ofparabolic partial differential equations in two or more dimensions. Typically, after a semidiscretization of the differential operator by the finite or boundary element method, we have a large statespace dimension n. In order to accelerate the simulation time or to facilitate the control design, it is often desirable to employ an approximate reducedorder system of order r, with r « n, instead of the original largescale system. We show how to compute a reducedorder system with a balancingrelated model reduction method. The method is based on the computation of the crossGramian X, which is the solution of a Sylvester equation. As standard algorithms for the solution of Sylvester equations are of limited use for largescale (possibly dense) systems, we investigate approaches based on the iterative sign function method, using datasparse matrix approximations (the hierarchical matrix format) and an approximate arithmetic. Furthermore, we use a modified iteration scheme for computing lowrank factors of the solution X. The projection matrices for MOR are computed from the dominant invariant subspace of X. We propose an efficient algorithm for the direct calculation of these projectors from the lowrank factors of X. Numerical experiments demonstrate the performance of the new approach. © 2008, Kent State University.