Low-rank methods for systems of Sylvester-type matrix equations

Low-rank approximation techniques are a very effective and widely used strategy for reducing the dimensionality of the data in knowledge-based applications such as machine learning and image processing. The aim of the project is to investigate the computation of low-rank approximations to the solution of systems of large-scale Sylvester-type matrix equations. Such systems arise in several applications in control theory, such as singular system control, robust control, and feedback control, research areas where high-dimensional data are all but uncommon. Building on recent advances in the field of matrix equations as well as on novel techniques in perturbation theory, we will identify different classes of systems that are suitable for specific solution strategies. We will characterize the systems for which methods based on the singular value decomposition are guaranteed to provide accurate solutions, and for the cases that require a new approach, we will develop algorithms for efficiently computing low-rank solutions that take into account structural properties of the original problem such as sparsity, rank, or prescribed eigenstructure of the matrix coefficients. In order to increase the efficiency of our methods and tackle even larger problems, we will resort to randomised projection techniques.