Riemannian Modified Polak--Ribière--Polyak Conjugate Gradient Order Reduced Model by Tensor Techniques

Abstract

This paper presents a new Riemannian modified Polak--Ribière--Polyak conjugate gradient algorithm to construct the reduced systems of quadratic-bilinear systems. We eliminate the orthogonality and homogeneity constraints of the truncated $\mathcal{H}_{2}$ optimal model order reduction problem and turn this constrained minimization problem into an unconstrained Riemannian optimization problem on the Grassmann manifold. Due to the compactness of this manifold, the existence of the minimum solution can be guaranteed. Applying tensor techniques, the Riemannian gradient of the cost function is derived. Additionally, we design a new Riemannian MPRP conjugate gradient scheme using the differentiated retraction and the scaled vector transport. The resulting search direction always provides a descent direction. The global convergence of the proposed algorithm is established. Moreover, our algorithm is also applicable to the minimization problems of linear and bilinear systems. Finally, two numerical tests are reported to illustrate the effectiveness of the proposed algorithm.