Parareal in time algorithm : an algebraic/virtual control presentation
Yvon Maday
The parareal in time algorithm is a recent iterative approach for using parallel platforms for the resolution of time dependant problems. One way to present it is through prediction correction paradigm where the corrections are solved independently, in parallel, from the use of a fine and accurate solver over each interval [Tn, Tn+1] where the time-instances T0=0<T1<..<T-N=T are properly chosen. The prediction is done through the use of a coarse solver, obtained by degradation of the fine solver that is used in serial and is much cheaper than the fine solver.
Another presentation, is done through virtual control theory as introduced by J.-L. Lions which allows for solving control problem in parallel in conjunction with other standard approaches. The idea there is to combine the two iterative processes (parareal ad optimal control) and get the coherent convergence of both processes.
This second presentation actually leads to propose an algebraic version of the parareal in time algorithm that can be explained as a preconditioner for solving the problem of finding the vector of all values of the solution at each time Tn.
We shall illustrate the presentation by many numerical simulations derived from the papers
J.-L. Lions, Y. Maday, and G. Turinici, A parareal in time discretization of pde's, C.R.
Acad. Sci. Paris, Serie I, 332 (2001), pp. 661{668.
Y. Maday, G. Turinici : A parareal in time procedure for the control of partial differential equations, C. R. Math. Acad. Sci. Paris 335 (2002), no 4, 387--392
Y. Maday, G. Turinici : Parallel in time algorithms for quantum control : the parareal time discretization scheme ; IJQC (2003).
Y. Maday, G. Turinici : New formulations of monotonically convergent quantum control algorithmes Jour. Of Chem. Phys. (2003) 118-17
Y. Maday, J. Salomon and G. Turinici. Discretely monotonically convergent algorithms in quantum control, IFAC LHMNLC03 Proceedings, Seville, 2003
P. Fischer, F. Hecht, Y. Maday , A parareal in time approximation of the Navier Stokes equations, Lecture Notes in Computational Science and Engineering , Vol. 40 Kornhuber, R.; Hoppe, R.; Périaux, J.; Pironneau, O.; Widlund, O.; Xu, J. (Eds.) 2004
Y. Maday, G. Turinici, Combining domain decomposition method with the parareal algorithm, Lecture Notes in Computational Science and Engineering , Vol. 40 Kornhuber, R.; Hoppe, R.; Périaux, J.; Pironneau, O.; Widlund, O.; Xu, J. (Eds.) 2004
Y. Maday, J. Salomon, G. Turinici: Monotonic time-discretized schemes in quantum control. Numer. Math. 103 (2006), no. 2, 323--338