Sketched and Truncated Polynomial Krylov Subspace Methods: Matrix Equations

Thanks to its great potential in reducing both computational cost and memory requirements, combining sketching and Krylov subspace techniques has attracted a lot of attention in the recent literature on projection methods for linear systems, matrix function approximations, and eigenvalue problems. Applying this appealing strategy in the context of linear matrix equations turns out to be far more involved than a straightforward generalization. These difficulties include establishing well-posedness of the projected problem and deriving possible error estimates depending on the sketching properties. Further computational complications include the lack of a natural residual norm estimate and of an explicit basis for the generated subspace. In this paper we propose a new sketched-and-truncated polynomial Krylov subspace method for Sylvester equations that aims to address all these issues. The potential of our novel approach, in terms of both computational time and storage demand, is illustrated with numerical experiments. Comparisons with a state-of-the-art projection scheme based on rational Krylov subspaces are also included.