Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347-360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385-391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191-8196]. In Maday et al. [Num. Math. (2006) 323-338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.
Mots-clés : quantum control, monotonic schemes, optimal control, Łojasiewicz inequality
@article{M2AN_2007__41_1_77_0,
author = {Salomon, Julien},
title = {Convergence of the time-discretized monotonic schemes},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {77--93},
publisher = {EDP-Sciences},
volume = {41},
number = {1},
year = {2007},
doi = {10.1051/m2an:2007008},
mrnumber = {2323691},
zbl = {1124.65059},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007008/}
}
TY - JOUR AU - Salomon, Julien TI - Convergence of the time-discretized monotonic schemes JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 77 EP - 93 VL - 41 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007008/ DO - 10.1051/m2an:2007008 LA - en ID - M2AN_2007__41_1_77_0 ER -
%0 Journal Article %A Salomon, Julien %T Convergence of the time-discretized monotonic schemes %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 77-93 %V 41 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007008/ %R 10.1051/m2an:2007008 %G en %F M2AN_2007__41_1_77_0
Salomon, Julien. Convergence of the time-discretized monotonic schemes. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 1, pp. 77-93. doi : 10.1051/m2an:2007008. http://www.numdam.org/articles/10.1051/m2an:2007008/
[1] and, Exponential split operator methods for solving coupled time-dependent Schrödinger equations. J. Chem. Phys. 99 (1993) 1185-1193.
[2] , Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851-956. | Zbl
[3] and, On the convergence of the proximal point algorithm for nonsmooth functions involving analytic features. Math. Program. (to appear). | MR
[4] and, Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena. J. Math. Chem. 31 (2002) 17-63. | Zbl
[5] , and, Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463-484. | Zbl
[6] and, Optimal bilinear control of an abstract Schrödinger equation. SIAM J. Cont. Opt. (to appear). | MR | Zbl
[7] and, Teaching lasers to control molecules. Phys. Rev. Lett 68 10 (1992) 1500-1503.
[8] , Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du CNRS, Les équations aux dérivées partielles 117 (1963). | Zbl
[9] , Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier 43 (1993) 1575-1595. | Numdam | Zbl
[10] and, New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys 118 18 (2003) 8191-8196.
[11] , and, Discretely monotonically convergent algorithm in quantum control, in Proc. LHMNLC03 IFAC conference, Sevilla (2003) 321-324.
[12] , and, Monotonic time-discretized schemes in quantum control. Num. Math. 103 (2006) 323-338. | Zbl
[13] , and, Control of quantum dynamics: Concepts, procedures and future prospects, in Computational Chemistry, Special Volume (C. Le Bris Editor) of Handbook of Numerical Analysis, Vol. X, edited by Ph.G. Ciarlet, Elsevier Science B.V. (2003). | MR | Zbl
[14] , Limit points of the monotonic schemes in quantum control, in Proc. 44th IEEE Conference on Decision and Control, Sevilla (2005).
[15] , and, Optimal control of selective vibrational excitation in harmonic linear chain molecules. J. Chem. Phys. 88 (1988) 6870-6883.
[16] , Accurate partial difference methods. I: Linear cauchy problems. Arch. Rat. Mech. An. 12 (1963) 392-402. | Zbl
[17] , Absorbing boundary conditions for nonlinear Schrödinger equation. Num. Math. 104 (2006) 103-127. | Zbl
[18] , and, Control of photochemical branching: Novel procedures for finding optimal pulses and global upper bounds, in Time Dependent Quantum Molecular Dynamics, J. Broeckhove, L. Lathouwers Eds., Plenum (1992) 347-360.
[19] ,,,,, and, A comparative study of time dependent quantum mechanical wave packet evolution methods. J. Chem. Phys. 96 (1992) 2077-2084.
[20] and, A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator. J. Chem. Phys. 109 (1998) 385-391.
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