Rational interpolation methods for symmetric Sylvester equations / Peter Benner, Tobias Breiten

Abstract

Abstract: We discuss low rank approximation methods for large-scale symmetric Sylvester equations. Following similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank nr, we derive necessary conditions for an approximation being optimal with respect to this norm. We show that the norm minimization problem is related to an objective function based on the H2-inner product for symmetric state space systems. This objective function is shown to exhibit first-order conditions that are equivalent to the ones from the norm minimization problem. We further propose an iterative procedure and demonstrate its efficiency when used within image reconstruction problems.