Model reduction is one of the most valuable contributions of mathematical system theory to applied mathematics (Jan C. Willems, 2007). In this book, Ulrike Baur investigates model reduction for parabolic PDEs based on balanced truncation. Effective implementations of balanced truncation (and of related methods such as singular perturbation approximation, cross-Gramian model reduction, balanced truncation for unstable systems) are developed by exploiting the structure of the underlying PDE problem. This is achieved by a data-sparse approximation of the (semi-) discretized differential operator using the hierarchical matrix format and by using the formatted arithmetic in the developed solvers for different types of matrix equations such as (generalized) Lyapunov, Stein, Sylvester and algebraic Bernoulli equations. All approaches are well-suited for classes of practically relevant problems and allow the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.

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