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

详细信息    查看全文

• 作者：Bernard Bonnard (1) (2)
  Mathieu Claeys (3) (4)
  Olivier Cots (2)
  Pierre Martinon (5)
  
  1. Institut de Math茅matiques de Bourgogne ; Universit茅 de Bourgogne ; 9 avenue Alain Savary ; 21078 ; Dijon ; France
  2. INRIA Sophia Antipolis M茅diterran茅e ; 06902 ; Sophia Antipolis ; France
  3. CNRS ; LAAS ; 7 avenue du colonel Roche ; 31077 ; Toulouse ; France
  4. Universit茅 de Toulouse ; UPS ; INSA ; INP ; ISAE ; UT1 ; UTM ; LAAS ; 31077 ; Toulouse ; France
  5. Inria and Ecole Polytechnique ; 91128 ; Palaiseau ; France
  
• 关键词：Geometric optimal control ; Contrast imaging in NMR ; Direct method ; Shooting and continuation techniques ; Moment optimization
• 刊名：Acta Applicandae Mathematicae
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：135
• 期：1
• 页码：5-45
• 全文大小：1,494 KB
• 参考文献：1. Allgower, E., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol.聽45. Soc. for Industrial and Applied Math, Philadelphia (2003), xxvi+388聽pp. 37/1.9780898719154" target="_blank" title="It opens in new window">CrossRef
  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鈥?1 (2001) 37/S0895479899358194" target="_blank" title="It opens in new window">CrossRef
  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鈥?59 (2013) CrossRef
  4. Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
  5. Bonnans, F.J., Martinon, P., Gr茅lard, V.: Bocop鈥攁聽collection of examples. Technical report RR-8053, INRIA (2012)
  6. Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathematics & Applications, vol.聽40. Springer, Berlin (2003), xvi+357聽pp.
  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鈥?12 (2012) 3500504" target="_blank" title="It opens in new window">CrossRef
  8. Bonnard, B., Caillau, J.-B., Tr茅lat, E.: Geometric optimal control of elliptic Keplerian orbits. Discrete Contin. Dyn. Syst., Ser. B 5(4), 929鈥?56 (2005) 3934/dcdsb.2005.5.929" target="_blank" title="It opens in new window">CrossRef
  9. Bonnard, B., Caillau, J.-B., Tr茅lat, E.: Second order optimality conditions in the smooth case and applications in optimal control. ESAIM Control Optim. Calc. Var. 13(2), 207鈥?36 (2007) CrossRef
  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鈥?969 (2012) CrossRef
  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鈥?32 (2013) 3934/mcrf.2013.3.397" target="_blank" title="It opens in new window">CrossRef
  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鈥?349 (2013) 37/110833427" target="_blank" title="It opens in new window">CrossRef
  13. Bulirsch, R., Stoer, J.: Introduction to Numerical Analysis, 2nd edn. Texts in Applied Mathematics, vol.聽12. Springer, New York (1993), xvi+744聽pp.
  14. Caillau, J.-B., Daoud, B.: Minimum time control of the circular restricted three-body problem. SIAM J. Control Optim. 50(6), 3178鈥?202 (2011) 37/110847299" target="_blank" title="It opens in new window">CrossRef
  15. Caillau, J.-B., Cots, O., Gergaud, J.: Differential continuation for regular optimal control problems. Optim. Methods Softw. 27(2), 177鈥?96 (2012) 3625" target="_blank" title="It opens in new window">CrossRef
  16. Chitour, Y., Jean, F., Tr茅lat, E.: Genericity results for singular curves. J. Differ. Geom. 73(1), 45鈥?3 (2006)
  17. Cots, O.: Contr么le optimal g茅om茅trique: m茅thodes homotopiques et applications. PhD thesis, Institut Math茅matiques de Bourgogne, Dijon, France (2012)
  18. Gebremedhin, A., Pothen, A., Walther, A.: Exploiting sparsity in Jacobian computation via coloring and automatic differentiation: a聽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.聽64, pp.聽327鈥?38. Springer, Berlin (2008)
  19. Gerdts, M.: Optimal Control of ODEs and DAEs. De Gruyter, Berlin (2011). 458 pages
  20. Hasco毛t, L., Pascual, V.: The Tapenade Automatic Differentiation tool: principles, model, and specification. Rapport de recherche RR-7957, INRIA (2012)
  21. Henrion, D., Lasserre, J.B., L枚fberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4鈥?), 761鈥?79 (2009) CrossRef
  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)
  23. Krener, A.J.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256鈥?93 (1977) 37/0315019" target="_blank" title="It opens in new window">CrossRef
  24. Kupka, I.: Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case. Trans. Am. Math. Soc. 299(1), 225鈥?43 (1987)
  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)
  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) 38/srep00589" target="_blank" title="It opens in new window">CrossRef
  27. Lasserre, J.B.: Positive Polynomials and Their Applications. Imperial College Press, London (2009) CrossRef
  28. Lasserre, J.B., Henrion, D., Prieur, C., Tr茅lat, E.: Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM J. Control Optim. 47(4), 1643鈥?666 (2008) 37/070685051" target="_blank" title="It opens in new window">CrossRef
  29. Levitt, M.H.: Spin Dynamics: Basics of Nuclear Magnetic Resonance. Wiley, New York (2001)
  30. Li Shin, Jr.: Control of inhomogeneous ensembles. PhD dissertation, Harvard University (2006)
  31. Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235鈥?57 (1976) 35706" target="_blank" title="It opens in new window">CrossRef
  32. Mor茅, J.J., Garbow, B.S., Hillstrom, K.E.: User guide for MINPACK-1, ANL-80-74, Argonne National Laboratory (1980)
  33. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) CrossRef
  34. Pontryagin, L.S., Boltyanski沫, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Matematicheskaya Teoriya Optimalnykh Protsessov, 4th edn. Nauka, Moscow (1983)
  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)
  36. W盲chter, 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鈥?7 (2006) CrossRef
  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)
  
• 刊物主题：Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9036

文摘

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.