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

A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs