  In this article, we derive fast and robust parallel-in-time preconditioned
iterative methods for the all-at-once linear systems arising upon
discretization of time-dependent PDEs. The discretization we employ is based on
a Runge--Kutta method in time, for which the development of parallel solvers is
an emerging research area in the literature of numerical methods for
time-dependent PDEs. By making use of classical theory of block matrices, one
is able to derive a preconditioner for the systems considered. The block
structure of the preconditioner allows for parallelism in the time variable, as
long as one is able to provide an optimal solver for the system of the stages
of the method. We thus propose a preconditioner for the latter system based on
a singular value decomposition (SVD) of the (real) Runge--Kutta matrix
$A_{\mathrm{RK}} = U \Sigma V^\top$. Supposing $A_{\mathrm{RK}}$ is invertible,
we prove that the spectrum of the system for the stages preconditioned by our
SVD-based preconditioner is contained within the right-half of the unit circle,
under suitable assumptions on the matrix $U^\top V$ (the assumptions are well
posed due to the polar decomposition of $A_{\mathrm{RK}}$). We show the
numerical efficiency of our SVD-based preconditioner by solving the system of
the stages arising from the discretization of the heat equation and the Stokes
equations, with sequential time-stepping. Finally, we provide numerical results
of the all-at-once approach for both problems, showing the speed-up achieved on
a parallel architecture.
