Speaker
Description
The research area of model reduction enables the acceleration of engineering design cycles through the construction of low-dimensional surrogate models. In this talk, we focus on the reduction of linear structured systems (LSS) from the perspective of balanced truncation. Such systems exhibit particular structures, which may involve second-order derivatives, time delays, or integro-differential operators. Balanced truncation eliminates states that are both hard to reach and hard to observe, based on the energy interpretation of the Gramians. However, computing the controllability and observability Gramians or their square-root factors can be challenging and computationally expensive. Here, we focus on their efficient computation for LSS. While for standard linear time-invariant systems, one can efficiently solve Lyapunov equations to compute the Gramians, there is no such general algebraic Lyapunov equation that encodes the Gramians for LSS. Instead, natural approaches consider the integral frequency domain representation to approximate the Gramians using quadrature rules. We identify these approximations with the solutions to generalized Sylvester equations, allowing us to employ and adapt an existing active sampling strategy and compute low-rank factors of the Gramians as solutions to the matrix equations. The procedure iteratively selects quadrature nodes to serve as interpolation points that provide the most relevant information while constructing low-rank solutions to the derived matrix equation. Thereby, each step involves solving a corresponding optimization problem in a low-dimensional subspace and the computation of the maximum residual error leading to the next quadrature node. In addition, we illustrate the proposed method for obtaining reduced-order models via balanced truncation on some numerical examples and compare it to standard methods for computing Gramians.
