Parallel in Time Algorithms for Quantum Control: Parareal Time Discretization Scheme

摘要

The numerical study of control problems in quantum chemistry requries many computer simulations of the dynamic phenomena involved. These simulationsa are, in many cases, too expensive to be carried out fro complex systems, thus precluding the treatment of interesting practical situations. In a context of fast increasing in both the CPU power available on typical workstationsa nd 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 begins to exceed the needs). In this context, the "parareal" time algorithm that parallelizes in the time direction the work required to solve the evolution equations has been introduced in previous works (Lions, J.-L.; Maday, Y.; Turinici, G.CR Acad Sci Paris I Math 2001, 332 (7), 661-668, Bal, G.; Maday, Y.In: Proceedings of the Workship on Domain Decomposition, LNCSE Series; Springer-Verlag: Berlin, 2001, 189-202). The theoretical modifications required to apply this algorithm to control problems and more specifically to the problems of quantum control are the main topic to this article. The preliminary results that are presented at the end of the article illustrate the feasibility of the approach and the potential for large time savings.