Krylov-based model reduction of second-order systems with proportional damping

In this note, we examine Krylov-based model reduction of second order systems where proportional damping is used to model energy dissipation. We give a detailed analysis of the distribution of system poles, and then, through a connection with potential theory, we are able to exploit the structure of these poles to obtain an optimal single shift strategy used in rational Krylov model reduction. We show that unlike the general case that requires usage of a second-order Krylov subspace structure, one can build up approximating subspaces satisfying all required conditions much more cheaply as direct sums of standard rational Krylov subspaces within the smaller component subspaces. Numerical examples are provided to illustrate and support the analysis.