  In 2008, Maday and Ronquist introduced an interesting new approach for the
direct parallel-in-time (PinT) solution of time-dependent PDEs. The idea is to
diagonalize the time stepping matrix, keeping the matrices for the space
discretization unchanged, and then to solve all time steps in parallel. Since
then, several variants appeared, and we call these closely related algorithms
ParaDiag algorithms. ParaDiagalgorithms in the literature can be classified
into two groups:
  ParaDiag-I: direct standalone solvers,
  ParaDiag-II: iterative solvers.
  We will explain the basic features of each group in this note. To have
concrete examples, we will introduce ParaDiag-I and ParaDiag-II for the
advection-diffusion equation. We will also introduce ParaDiag-II for the wave
equation and an optimal control problem for the wave equation. We could have
used the advection-diffusion equation as well to illustrate ParaDiag-II, but
wave equations are known to cause problems for certain PinT algorithms and thus
constitute an especially interesting example for which ParaDiag algorithms were
tested. We show the main known theoretical results in each case, and also
provide Matlab codes for testing. The goal of the Matlab codes is to help the
interested reader understand the key features of the ParaDiag algorithms,
without intention to be highly tuned for efficiency and/or low memory use.
  We also provide speedup measurements of ParaDiag algorithms for a 2D linear
advection-diffusion equation. These results are obtained on the Tianhe-1
supercomputer in China and the SIUE Campus Cluster in the US and and we compare
these results to the performance of parareal and MGRiT, two widely used PinT
algorithms.
