Acta Applicandae Mathematicae

, Volume 135, Issue 1, pp 5–45 | Cite as

Geometric and Numerical Methods in the Contrast Imaging Problem in Nuclear Magnetic Resonance

  • Bernard Bonnard
  • Mathieu Claeys
  • Olivier Cots
  • Pierre Martinon
Article

Abstract

In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. This leads to a numerical investigation based on so-called indirect methods using the HamPath software. The results are then compared with a direct method implemented within the Bocop toolbox. Finally lmi techniques are used to estimate a global optimum.

Keywords

Geometric optimal control Contrast imaging in NMR Direct method Shooting and continuation techniques Moment optimization 

References

  1. 1.
    Allgower, E., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol. 45. Soc. for Industrial and Applied Math, Philadelphia (2003), xxvi+388 pp. CrossRefMATHGoogle Scholar
  2. 2.
    Amestoy, P.R., Duff, I.S., Koster, J., Excellent, J.-Y.L.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Aronna, M.S., Bonnans, F.J., Martinon, P.: A shooting algorithm for optimal control problems with singular arcs. J. Optim. Theory Appl. 158(2), 419–459 (2013) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001) MATHGoogle Scholar
  5. 5.
    Bonnans, F.J., Martinon, P., Grélard, V.: Bocop—a collection of examples. Technical report RR-8053, INRIA (2012) Google Scholar
  6. 6.
    Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathematics & Applications, vol. 40. Springer, Berlin (2003), xvi+357 pp. MATHGoogle Scholar
  7. 7.
    Bonnard, B., Cots, O.: Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance. Math. Models Methods Appl. Sci. 24(1), 187–212 (2012) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bonnard, B., Caillau, J.-B., Trélat, E.: Geometric optimal control of elliptic Keplerian orbits. Discrete Contin. Dyn. Syst., Ser. B 5(4), 929–956 (2005) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bonnard, B., Caillau, J.-B., Trélat, E.: Second order optimality conditions in the smooth case and applications in optimal control. ESAIM Control Optim. Calc. Var. 13(2), 207–236 (2007) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bonnard, B., Cots, O., Glaser, S., Lapert, M., Sugny, D., Zhang, Y.: Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance. IEEE Trans. Autom. Control 57(8), 1957–1969 (2012) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bonnard, B., Chyba, M., Jacquemard, A., Marriott, J.: Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Math. Control Relat. Fields 3(4), 397–432 (2013) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Bonnard, B., Chyba, M., Marriott, J.: Singular trajectories and the contrast imaging problem in nuclear magnetic resonance. SIAM J. Control Optim. 51(2), 1325–1349 (2013) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Bulirsch, R., Stoer, J.: Introduction to Numerical Analysis, 2nd edn. Texts in Applied Mathematics, vol. 12. Springer, New York (1993), xvi+744 pp. MATHGoogle Scholar
  14. 14.
    Caillau, J.-B., Daoud, B.: Minimum time control of the circular restricted three-body problem. SIAM J. Control Optim. 50(6), 3178–3202 (2011) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Caillau, J.-B., Cots, O., Gergaud, J.: Differential continuation for regular optimal control problems. Optim. Methods Softw. 27(2), 177–196 (2012) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Chitour, Y., Jean, F., Trélat, E.: Genericity results for singular curves. J. Differ. Geom. 73(1), 45–73 (2006) MATHGoogle Scholar
  17. 17.
    Cots, O.: Contrôle optimal géométrique: méthodes homotopiques et applications. PhD thesis, Institut Mathématiques de Bourgogne, Dijon, France (2012) Google Scholar
  18. 18.
    Gebremedhin, A., Pothen, A., Walther, A.: Exploiting sparsity in Jacobian computation via coloring and automatic differentiation: a case study in a simulated moving bed process. In: Bischof, C., et al. (eds.) Proceedings of the Fifth International Conference on Automatic Differentiation (AD2008). Lecture Notes in Computational Science and Engineering, vol. 64, pp. 327–338. Springer, Berlin (2008) Google Scholar
  19. 19.
    Gerdts, M.: Optimal Control of ODEs and DAEs. De Gruyter, Berlin (2011). 458 pages Google Scholar
  20. 20.
    Hascoët, L., Pascual, V.: The Tapenade Automatic Differentiation tool: principles, model, and specification. Rapport de recherche RR-7957, INRIA (2012) Google Scholar
  21. 21.
    Henrion, D., Lasserre, J.B., Löfberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Henrion, D., Daafouz, J., Claeys, M.: Optimal switching control design for polynomial systems: an LMI approach. In: Proceedings of the 52rd IEEE Conf. Decision and Control, CDC 2013, Firenze, Italy (2013) Google Scholar
  23. 23.
    Krener, A.J.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256–293 (1977) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kupka, I.: Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case. Trans. Am. Math. Soc. 299(1), 225–243 (1987) MATHMathSciNetGoogle Scholar
  25. 25.
    Lapert, M., Zhang, Y., Glaser, S.J., Sugny, D.: Towards the time-optimal control of dissipative spin-1/2 particles in nuclear magnetic resonance. J. Phys. B, At. Mol. Opt. Phys. 44(15) (2011) Google Scholar
  26. 26.
    Lapert, M., Zhang, Y., Janich, M., Glaser, S.J., Sugny, D.: Exploring the physical limits of saturation contrast in Magnetic Resonance Imaging. Sci. Rep. 2, 589 (2012) CrossRefGoogle Scholar
  27. 27.
    Lasserre, J.B.: Positive Polynomials and Their Applications. Imperial College Press, London (2009) CrossRefGoogle Scholar
  28. 28.
    Lasserre, J.B., Henrion, D., Prieur, C., Trélat, E.: Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM J. Control Optim. 47(4), 1643–1666 (2008) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Levitt, M.H.: Spin Dynamics: Basics of Nuclear Magnetic Resonance. Wiley, New York (2001) Google Scholar
  30. 30.
    Li Shin, Jr.: Control of inhomogeneous ensembles. PhD dissertation, Harvard University (2006) Google Scholar
  31. 31.
    Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235–257 (1976) CrossRefMATHGoogle Scholar
  32. 32.
    Moré, J.J., Garbow, B.S., Hillstrom, K.E.: User guide for MINPACK-1, ANL-80-74, Argonne National Laboratory (1980) Google Scholar
  33. 33.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) CrossRefMATHGoogle Scholar
  34. 34.
    Pontryagin, L.S., Boltyanskiĭ, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Matematicheskaya Teoriya Optimalnykh Protsessov, 4th edn. Nauka, Moscow (1983) Google Scholar
  35. 35.
    Powell, M.J.D.: A hybrid method for nonlinear equations. In: Rabinowitz, P. (ed.) Numerical Methods for Nonlinear Algebraic Equations. Gordon & Breach, New York (1970) Google Scholar
  36. 36.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Walther, A., Griewank, A.: Getting started with adol-c. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing. Chapman-Hall/CRC Computational Science, London (2012) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Bernard Bonnard
    • 1
    • 2
  • Mathieu Claeys
    • 3
    • 4
  • Olivier Cots
    • 2
  • Pierre Martinon
    • 5
  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijonFrance
  2. 2.INRIA Sophia Antipolis MéditerranéeSophia AntipolisFrance
  3. 3.CNRSLAASToulouseFrance
  4. 4.Université de Toulouse, UPS, INSA, INP, ISAEUT1, UTM, LAASToulouseFrance
  5. 5.Inria and Ecole PolytechniquePalaiseauFrance

Personalised recommendations