Hostname: page-component-7dd5485656-dk7s8 Total loading time: 0 Render date: 2025-10-30T02:33:26.183Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of the time-discretized monotonic schemes

Published online by Cambridge University Press:  26 April 2007

Julien Salomon
Julien Salomon*
Affiliation:
Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret 75013 Paris, France. Université Paris-Dauphine, Paris 9, CEREMADE, Place du Maréchal Lattre de Tassigny, 75775 Paris Cedex 16, France. Julien.Salomon@dauphine.fr
Get access

Abstract

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced byTannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or theirunified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. InMaday et al. [Num. Math. (2006) 323–338], a time discretization which preserves theproperty of monotonicity has been presented. This paper introduces aproof of the convergence of these schemes and some results regarding theirrate of convergence.

Keywords

Quantum controlmonotonic schemesoptimal controlŁojasiewicz inequality.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 77 - 93
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bandrauk, A.D. and Shen, H., Exponential split operator methods for solving coupled time-dependent Schrödinger equations. J. Chem. Phys. 99 (1993) 1185–1193. CrossRef
Beauchard, K., Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851–956. CrossRef
J. Bolte and H. Attouch, On the convergence of the proximal point algorithm for nonsmooth functions involving analytic features. Math. Program. (to appear).
Brown, E. and Rabitz, H., Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena. J. Math. Chem. 31 (2002) 17–63. CrossRef
Haraux, A., Jendoubi, M.A. and Kavian, O., Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463–484. CrossRef
K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation. SIAM J. Cont. Opt. (to appear).
R. Judson and H. Rabitz, Teaching lasers to control molecules. Phys. Rev. Lett 68 10 (1992) 1500–1503.
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du CNRS, Les équations aux dérivées partielles 117 (1963).
Łojasiewicz, S., Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier 43 (1993) 1575–1595. CrossRef
Y. Maday and G. Turinici, New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys 118 18 (2003) 8191–8196.
Y. Maday, J. Salomon and G. Turinici, Discretely monotonically convergent algorithm in quantum control, in Proc. LHMNLC03 IFAC conference , Sevilla (2003) 321–324.
Maday, Y., Salomon, J. and Turinici, G., Monotonic time-discretized schemes in quantum control. Num. Math. 103 (2006) 323–338. CrossRef
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).
J. Salomon, Limit points of the monotonic schemes in quantum control, in Proc. 44th IEEE Conference on Decision and Control , Sevilla (2005).
Shi, S., Woody, A. and Rabitz, H., Optimal control of selective vibrational excitation in harmonic linear chain molecules. J. Chem. Phys. 88 (1988) 6870–6883. CrossRef
Strang, G., Accurate partial difference methods. I: Linear cauchy problems. Arch. Rat. Mech. An. 12 (1963) 392–402. CrossRef
Szeftel, J., Absorbing boundary conditions for nonlinear Schrödinger equation. Num. Math. 104 (2006) 103–127. CrossRef
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.
Truong, T.N., Tanner, J.J., Bala, P., McCammon, J.A., Kouri, D.J., Lesyng, B. and Hoffman, D.K., A comparative study of time dependent quantum mechanical wave packet evolution methods. J. Chem. Phys. 96 (1992) 2077–2084.
Zhu, W. and Rabitz, H., 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. CrossRef