Riemannian optimization approach to structure-preserving model order reduction of integral-differential systems on the product of two Stiefel manifolds

Abstract

In this paper, we discuss the structure-preserving H2 optimal model order reduction (MOR) problem of integral-differential systems based on the projection technique. Different from the common idea, we take a new way to deal with this minimization problem and for the first time reduce the integral-differential systems on the product of two Stiefel manifolds. The cost function is viewed as a function whose parameters are a pair of projection matrices with different dimensions, and then the H2 optimal MOR problem is reformulated as the minimization problem of the product of two Stiefel manifolds. We extend the Riemannian geometry of the single Stiefel manifold to the product manifold and the Riemannian gradient of the cost function is derived using the orthogonal projection of the product manifold. Further, we design a Riemannian conjugate gradient scheme on the product manifold, and propose a structure-preserving Riemannian conjugate gradient algorithm. The resulting search direction is a descent direction of the cost function. Besides, we show that our algorithm is globally convergent and has the potential application to second-order systems. Finally, numerical examples demonstrate the effectiveness of the proposed algorithm.