×

zbMATH — the first resource for mathematics

Geometric and numerical methods in the contrast imaging problem in nuclear magnetic resonance. (English) Zbl 1311.49065
Summary: 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, 1mi techniques are used to estimate a global optimum.

MSC:
49M05 Numerical methods based on necessary conditions
49K15 Optimality conditions for problems involving ordinary differential equations
49N90 Applications of optimal control and differential games
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 1036.65047
[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, 15-41, (2001) · Zbl 0992.65018
[3] Aronna, M.S.; Bonnans, F.J.; Martinon, P., A shooting algorithm for optimal control problems with singular arcs, J. Optim. Theory Appl., 158, 419-459, (2013) · Zbl 1275.49045
[4] Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001) · Zbl 0995.49017
[5] Bonnans, F.J., Martinon, P., Grélard, V.: Bocop—a collection of examples. Technical report RR-8053, INRIA (2012) · Zbl 1268.49023
[6] Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathematics & Applications, vol. 40. Springer, Berlin (2003), xvi+357 pp. · Zbl 1022.93003
[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, 187-212, (2012) · Zbl 1281.49025
[8] Bonnard, B.; Caillau, J.-B.; Trélat, E., Geometric optimal control of elliptic Keplerian orbits, Discrete Contin. Dyn. Syst., Ser. B, 5, 929-956, (2005) · Zbl 1082.70014
[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, 207-236, (2007) · Zbl 1123.49014
[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, 1957-1969, (2012) · Zbl 1369.49028
[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, 397-432, (2013) · Zbl 1273.94128
[12] Bonnard, B.; Chyba, M.; Marriott, J., Singular trajectories and the contrast imaging problem in nuclear magnetic resonance, SIAM J. Control Optim., 51, 1325-1349, (2013) · Zbl 1268.49022
[13] Bulirsch, R., Stoer, J.: Introduction to Numerical Analysis, 2nd edn. Texts in Applied Mathematics, vol. 12. Springer, New York (1993), xvi+744 pp. · Zbl 0771.65002
[14] Caillau, J.-B.; Daoud, B., Minimum time control of the circular restricted three-body problem, SIAM J. Control Optim., 50, 3178-3202, (2011) · Zbl 1268.49023
[15] Caillau, J.-B.; Cots, O.; Gergaud, J., Differential continuation for regular optimal control problems, Optim. Methods Softw., 27, 177-196, (2012) · Zbl 1248.49025
[16] Chitour, Y.; Jean, F.; Trélat, E., Genericity results for singular curves, J. Differ. Geom., 73, 45-73, (2006) · Zbl 1102.53019
[17] Cots, O.: Contrôle optimal géométrique: méthodes homotopiques et applications. PhD thesis, Institut Mathématiques de Bourgogne, Dijon, France (2012) · Zbl 1178.90277
[18] Gebremedhin, A.; Pothen, A.; Walther, A.; Bischof, C. (ed.); etal., Exploiting sparsity in Jacobian computation via coloring and automatic differentiation: a case study in a simulated moving bed process, No. 64, 327-338, (2008), Berlin · Zbl 1152.65416
[19] Gerdts, M.: Optimal Control of ODEs and DAEs. De Gruyter, Berlin (2011). 458 pages · Zbl 1275.49001
[20] Hascoët, L., Pascual, V.: The Tapenade Automatic Differentiation tool: principles, model, and specification. Rapport de recherche RR-7957, INRIA (2012) · Zbl 1248.49025
[21] Henrion, D.; Lasserre, J.B.; Löfberg, J., Gloptipoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw., 24, 761-779, (2009) · Zbl 1178.90277
[22] Henrion, D.; Daafouz, J.; Claeys, M., Optimal switching control design for polynomial systems: an LMI approach, CDC, Firenze, Italy, 2013
[23] Krener, A.J., The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15, 256-293, (1977) · Zbl 0354.49008
[24] Kupka, I., Geometric theory of extremals in optimal control problems. I. the fold and Maxwell case, Trans. Am. Math. Soc., 299, 225-243, (1987) · Zbl 0606.49016
[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) · Zbl 1102.53019
[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)
[27] Lasserre, J.B.: Positive Polynomials and Their Applications. Imperial College Press, London (2009)
[28] Lasserre, J.B.; Henrion, D.; Prieur, C.; Trélat, E., Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47, 1643-1666, (2008) · Zbl 1188.90193
[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) · Zbl 1369.49044
[31] Maurer, H., Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl., 18, 235-257, (1976) · Zbl 0302.65063
[32] Moré, 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) · Zbl 0930.65067
[34] Pontryagin, L.S., Boltyanskiĭ, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Matematicheskaya Teoriya Optimalnykh Protsessov, 4th edn. Nauka, Moscow (1983)
[35] Powell, M.J.D.; Rabinowitz, P. (ed.), A hybrid method for nonlinear equations, (1970), New York
[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, 25-57, (2006) · Zbl 1134.90542
[37] Walther, A.; Griewank, A.; Naumann, U. (ed.); Schenk, O. (ed.), Getting started with adol-c, (2012), London
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.