Convergence of the time-discretized monotonic schemes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 77-93

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.

DOI : https://doi.org/10.1051/m2an:2007008
Classification:  49J20,  68W40
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     pages = {77-93},
     doi = {10.1051/m2an:2007008},
     zbl = {1124.65059},
     mrnumber = {2323691},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_1_77_0}
}
Salomon, Julien. Convergence of the time-discretized monotonic schemes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 77-93. doi : 10.1051/m2an:2007008. http://www.numdam.org/item/M2AN_2007__41_1_77_0/

[1] A.D. Bandrauk and H. Shen, Exponential split operator methods for solving coupled time-dependent Schrödinger equations. J. Chem. Phys. 99 (1993) 1185-1193.

[2] K. Beauchard, Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851-956. | Zbl 1124.93009

[3] J. Bolte and H. Attouch, On the convergence of the proximal point algorithm for nonsmooth functions involving analytic features. Math. Program. (to appear). | MR 2421270

[4] E. Brown and H. Rabitz, Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena. J. Math. Chem. 31 (2002) 17-63. | Zbl 0996.81001

[5] A. Haraux, M.A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463-484. | Zbl 1036.35035

[6] K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation. SIAM J. Cont. Opt. (to appear). | MR 2299629 | Zbl 1136.35089

[7] R. Judson and H. Rabitz, Teaching lasers to control molecules. Phys. Rev. Lett 68 10 (1992) 1500-1503.

[8] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du CNRS, Les équations aux dérivées partielles 117 (1963). | Zbl 0234.57007

[9] S. Łojasiewicz, Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier 43 (1993) 1575-1595. | Numdam | Zbl 0803.32002

[10] Y. Maday and G. Turinici, New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys 118 18 (2003) 8191-8196.

[11] Y. Maday, J. Salomon and G. Turinici, Discretely monotonically convergent algorithm in quantum control, in Proc. LHMNLC03 IFAC conference, Sevilla (2003) 321-324.

[12] Y. Maday, J. Salomon and G. Turinici, Monotonic time-discretized schemes in quantum control. Num. Math. 103 (2006) 323-338. | Zbl 1095.65058

[13] H. Rabitz, G. Turinici and E. Brown, 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 2008399 | Zbl 1066.81015

[14] J. Salomon, Limit points of the monotonic schemes in quantum control, in Proc. 44th IEEE Conference on Decision and Control, Sevilla (2005).

[15] S. Shi, A. Woody and H. Rabitz, Optimal control of selective vibrational excitation in harmonic linear chain molecules. J. Chem. Phys. 88 (1988) 6870-6883.

[16] G. Strang, Accurate partial difference methods. I: Linear cauchy problems. Arch. Rat. Mech. An. 12 (1963) 392-402. | Zbl 0113.32303

[17] J. Szeftel, Absorbing boundary conditions for nonlinear Schrödinger equation. Num. Math. 104 (2006) 103-127. | Zbl 1130.35119

[18] D. Tannor, V. Kazakov and V. Orlov, 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] T.N. Truong, J.J. Tanner, P. Bala, J.A. Mccammon, D.J. Kouri, B. Lesyng and D.K. Hoffman, A comparative study of time dependent quantum mechanical wave packet evolution methods. J. Chem. Phys. 96 (1992) 2077-2084.

[20] W. Zhu and H. Rabitz, 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.