A parallel in time approach for quantum control: the parareal algorithm

The numerical studies of control problems in quantum chemistry go through the computer simulation of the dynamical phenomena involved. These simulations are in many cases too expensive to be carried out for complex systems, precluding thus treatment of interesting practical situations. In a context of fast increasing in both the CPU power available on typical workstations and of the number of computers that can be connected through high speed networks, the difficulty lies rather in how to obtain "real time" solutions than in the amount of CPU power available (which becomes to exceed the needs). In this context, the "parareal" time algorithm that parallelize in the time direction the effort required to solve evolution equations has been introduced in previous works (Bal and Maday, 2001, and Lions et al., 2001). The theoretical modifications required in order to apply this algorithm to quantum control problems together with numerical evidence is the main topic of this paper.