  In this paper we propose to use model reduction techniques for speeding up
the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for
iteratively solving all-at-once systems from evolutionary PDEs. In particular,
we use the reduced basis method to seek a low-dimensional approximation to the
sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG
preconditioning procedure. Different from the standard reduced order modeling
that uses the separation of offline and online stages, we have to build the
reduced order model (ROM) online for the considered systems at each iteration.
Therefore, several heuristic acceleration techniques are introduced in the
greedy basis generation algorithm, that is built upon a residual-based error
indicator, to further boost up its computational efficiency. Several numerical
experiments are conducted, which illustrate the favorable computational
efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in
comparison with the state of the art multigrid-based ParaDIAG preconditioner.
