  We present and analyze a parallel implementation of a parallel-in-time method
based on $\alpha$-circulant preconditioned Richardson iterations. While there
are a lot of papers exploring this new class of single-level, time-parallel
integrators from many perspectives, performance results of actual parallel runs
are still missing. This leaves a critical gap, because the efficiency and
applicability heavily rely on the actual parallel performance, with only
limited guidance from theoretical considerations. Also, many challenges like
selecting good parameters, finding suitable communication strategies, and
performing a fair comparison to sequential time-stepping methods can be easily
missed. In this paper, we first extend the original idea by using a collocation
method of arbitrary order, which adds another level of parallelization in time.
We derive an adaptive strategy to select a new $\alpha$-circulant
preconditioner for each iteration during runtime for balancing convergence
rates, round-off errors and inexactness in the individual time-steps. After
addressing these more theoretical challenges, we present an open-source space-
and doubly-time-parallel implementation and evaluate its performance for two
different test problems.
