  In this paper, we consider the problem of accelerating the numerical
simulation of time dependent problems by time domain decomposition. The
available algorithms enabling such decompositions present severe efficiency
limitations and are an obstacle for the solution of large scale and high
dimensional problems. Our main contribution is the improvement of the parallel
efficiency of the parareal in time method. The parareal method is based on
combining predictions made by a numerically inexpensive solver (with coarse
physics and/or coarse resolution) with corrections coming from an expensive
solver (with high-fidelity physics and high resolution). At convergence, the
algorithm provides a solution that has the fine solver's high-fidelity physics
and high resolution. In the classical version, the fine solver has a fixed high
accuracy which is the major obstacle to achieve a competitive parallel
efficiency. In this paper, we develop an adaptive variant that overcomes this
obstacle by dynamically increasing the accuracy of the fine solver across the
parareal iterations. We theoretically show that the parallel efficiency becomes
very competitive in the ideal case where the cost of the coarse solver is
small, thus proving that the only remaining factors impeding full scalability
become the cost of the coarse solver and communication time. The developed
theory has also the merit of setting a general framework to understand the
success of several extensions of parareal based on iteratively improving the
quality of the fine solver and re-using information from previous parareal
steps. We illustrate the actual performance of the method in stiff ODEs, which
are a challenging family of problems since the only mechanism for adaptivity is
time and efficiency is affected by the cost of the coarse solver.
