Riemannian optimal model reduction of linear second-order systems

Abstract

This letter develops a structure preserving ${H^{2}}$ optimal model reduction method of linear second-order systems. The model reduction problem is formulated as an optimization problem on the product manifold of the two manifolds of symmetric positive definite matrices and two Euclidean manifolds. Reduced systems constructed by the optimal solution preserve the structure of the original second-order system. A Riemannian metric of the manifold is chosen in such a way that the manifold is geodesically complete, i.e., the domain of the exponential map is the whole tangent space for all points on the manifold. The Riemannian gradient of the objective function is derived for solving the problem by using a Riemannian steepest descent method. The geodesic completeness of the manifold guarantees that all points generated by the steepest descent method are on the manifold, and thus, our method naturally preserves the second-order structure. Furthermore, we suggest how to choose an initial point in the proposed algorithm. The initial point is given by solving another optimization problem on the Stiefel manifold. As a result, we can expect that the value of the objective function at the initial point is relatively small. Numerical experiments illustrate that the proposed method can give a reduced system which is sufficiently close to the original system even if the dimension of the reduced system is small.