Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.

Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.