# Convergence of the time-discretized monotonic schemes

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 1, page 77-93
- ISSN: 0764-583X

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topSalomon, Julien. "Convergence of the time-discretized monotonic schemes." ESAIM: Mathematical Modelling and Numerical Analysis 41.1 (2007): 77-93. <http://eudml.org/doc/250026>.

@article{Salomon2007,

abstract = {
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.
},

author = {Salomon, Julien},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Quantum control; monotonic schemes; optimal control; Ł
ojasiewicz inequality.; quantum control; Lojasiewicz inequality; time discretization; Schrödinger equation; convergence},

language = {eng},

month = {4},

number = {1},

pages = {77-93},

publisher = {EDP Sciences},

title = {Convergence of the time-discretized monotonic schemes},

url = {http://eudml.org/doc/250026},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Salomon, Julien

TI - Convergence of the time-discretized monotonic schemes

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/4//

PB - EDP Sciences

VL - 41

IS - 1

SP - 77

EP - 93

AB -
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.

LA - eng

KW - Quantum control; monotonic schemes; optimal control; Ł
ojasiewicz inequality.; quantum control; Lojasiewicz inequality; time discretization; Schrödinger equation; convergence

UR - http://eudml.org/doc/250026

ER -

## References

top- A.D. Bandrauk and H. Shen, Exponential split operator methods for solving coupled time-dependent Schrödinger equations. J. Chem. Phys.99 (1993) 1185–1193.
- K. Beauchard, Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl.84 (2005) 851–956. Zbl1124.93009
- J. Bolte and H. Attouch, On the convergence of the proximal point algorithm for nonsmooth functions involving analytic features. Math. Program. (to appear). Zbl1165.90018
- E. Brown and H. Rabitz, Some mathematical and algorithmic challenges in the control of quantum dynamics phenomena. J. Math. Chem.31 (2002) 17–63. Zbl0996.81001
- 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. Zbl1036.35035
- K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation. SIAM J. Cont. Opt. (to appear). Zbl1136.35089
- R. Judson and H. Rabitz, Teaching lasers to control molecules. Phys. Rev. Lett68 10 (1992) 1500–1503.
- S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du CNRS, Les équations aux dérivées partielles117 (1963).
- S. Łojasiewicz, Sur la géométrie semi- et sous-analytique. Ann. Inst. Fourier43 (1993) 1575–1595. Zbl0803.32002
- Y. Maday and G. Turinici, New formulations of monotonically convergent quantum control algorithms. J. Chem. Phys118 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.
- Y. Maday, J. Salomon and G. Turinici, Monotonic time-discretized schemes in quantum control. Num. Math.103 (2006) 323–338. Zbl1095.65058
- 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).
- 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.
- G. Strang, Accurate partial difference methods. I: Linear cauchy problems. Arch. Rat. Mech. An.12 (1963) 392–402. Zbl0113.32303
- J. Szeftel, Absorbing boundary conditions for nonlinear Schrödinger equation. Num. Math.104 (2006) 103–127.
- 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.
- 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.
- 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.

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