  Of all the possible projection methods for solving large-scale Lyapunov
matrix equations, Galerkin approaches remain much more popular than
minimal-residual ones. This is mainly due to the different nature of the
projected problems stemming from these two families of methods. While a
Galerkin approach leads to the solution of a low-dimensional matrix equation
per iteration, a matrix least-squares problem needs to be solved per iteration
in a minimal-residual setting. The significant computational cost of these
least-squares problems has steered researchers towards Galerkin methods in
spite of the appealing properties of minimal-residual schemes. In this paper we
introduce a framework that allows for modifying the Galerkin approach by
low-rank, additive corrections to the projected matrix equation problem with
the two-fold goal of attaining monotonic convergence rates similar to those of
minimal-residual schemes while maintaining essentially the same computational
cost of the original Galerkin method. We analyze the well-posedness of our
framework and determine possible scenarios where we expect the residual norm
attained by two low-rank-modified variants to behave similarly to the one
computed by a minimal-residual technique. A panel of diverse numerical examples
shows the behavior and potential of our new approach.
