Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

In this article, we deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters. Hence, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems we turn our focus to systems with a certain structure, for which the method can be applied particularly efficiently. We illustrate the effectiveness of our approach on several models from fluid dynamics and mechanics. We further consider an application of the method in the context of damping optimization