Comparing Loewner and Krylov based model order reduction for time delay systems

We use model order reduction to set up a reduced, linear model for high dimensional, time-delay problems. Three approaches are compared with respect to computational efficiency for setting up the reduced model, and, with respect to accuracy. The first approach uses a linear pencil formulation that is spectrally equivalent to a rational matrix-valued interpolant of the non-linear time delay formulation in the Laplace domain. The second approach is based on the Loewner framework. The Loewner framework is easier to use and is suitable for a black box problem. However, for high dimensional, white box models, the linear pencil formulation outperforms the Loewner framework. This is done by exploiting the structure of the problem and making use of a rational Krylov method with only a few distinct interpolation points, and, consequently, a small number of expensive sparse LU factorizations. In contrast, setting up the Loewner matrices requires many large scale linear system solves which, as our experiments show, can be the most important factor in the computation time. The third approach is a combination of both ideas, where the computationally more expensive Loewner approach is applied to the reduced model obtained by the Krylov approach.