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2014 | 24 | 2 | 249-257
Tytuł artykułu

Tikhonov regularization and constrained quadratic programming for magnetic coil design problems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this work, the problem of coil design is studied. It is assumed that the structure of the coil is known (i.e., the positions of simple circular coils are fixed) and the problem is to find current distribution to obtain the required magnetic field in a given region. The unconstrained version of the problem (arbitrary currents are allowed) can be formulated as a Least-SQuares (LSQ) problem. However, the results obtained by solving the LSQ problem are usually useless from the application point of view. Moreover, for higher dimensions the problem is ill-conditioned. To overcome these difficulties, a regularization term is sometimes added to the cost function, in order to make the solution smoother. The regularization technique, however, produces suboptimal solutions. In this work, we propose to solve the problem under study using the constrained Quadratic Programming (QP) method. The methods are compared in terms of the quality of the magnetic field obtained, and the power of the designed coil. Several 1D and 2D examples are considered. It is shown that for the same value of the maximum current the QP method provides solutions with a higher quality magnetic field than the regularization method.
Rocznik
Tom
24
Numer
2
Strony
249-257
Opis fizyczny
Daty
wydano
2014
otrzymano
2013-01-20
poprawiono
2013-08-18
Twórcy
  • Department of Electrical Engineering, AGH University of Science and Technology, al. Mickiewicza 30, 31-231 Cracow, Poland
  • Department of Electrical Engineering, AGH University of Science and Technology, al. Mickiewicza 30, 31-231 Cracow, Poland
Bibliografia
  • Bro, R. and Jong, S.D. (1997). A fast non-negativityconstrained least squares algorithm, Journal of Chemometrics 11(5): 393-401.
  • Fisher, B.J., Dillon, N., Carpenter, T.A. and Hall, L.D. (1997). Design of a biplanar gradient coil using a genetic algorithm, Magnetic Resonance Imaging 15(3): 369-376.
  • Garda, B. (2012). Linear algebra approach and the quasi-Newton algorithm for the optimal coil design problem, Przegląd Elektrotechniczny (7a): 261-264.
  • Garda, B. and Galias, Z. (2010). Comparison of the linear algebra approach and the evolutionary computing for magnetic field shaping in linear coils, Nonlinear Theory and Its Applications, NOLTA 2010, Cracow, Poland, pp. 508-511.
  • Garda, B. and Galias, Z. (2012). Non-negative least squares and the Tikhonov regularization methods for coil design problems, Proceedings of the International Conference on Signals and Electronic Systems, ICSES 2012, Wrocław, Poland.
  • Hansen, P. (1998). Regularization Tools: A Matlab Package for Analysis and Solution of Discrete Ill-posed Problems. Version 3.0 for Matlab 5.2, IMM-REP, Institut for Matematisk Modellering, Danmarks Tekniske Universitet, Kongens Lyngby.
  • Jin, J. (1999). Electromagnetic Analysis and Design in Magnetic Resonance Imaging, Biomedical Engineering Series, CRC Press, Boca Raton, FL.
  • Lawson, C. and Hanson, R. (1987). Solving Least Squares Problems, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • Macovski, A., Xu, H., Conolly, S. and Scott, G. (2000). Homogeneous magnet design using linear programing, IEEE Transactions on Magnetics 36(2): 476-483.
  • Prasath, V.B.S. (2011). A well-posed multiscale regularization scheme for digital image denoising, International Journal of Applied Mathematics and Computer Science 21(4): 769-777, DOI: 10.2478/v10006-011-0061-7.
  • Sikora, R., Krasoń, P. and Gramz, M. (1980). Magnetic field synthesis at the plane perpendicular to the axis of solenoid, Archiv fur elektrotechnik 62(3): 135-156.
  • Szynkiewicz, W. and Błaszczyk, J. (2011). Optimization-based approach to path planning for closed chain robot systems, International Journal of Applied Mathematics and Computer Science 21(4): 659-670, DOI: 10.2478/v10006-0110052-8.
  • Tikhonov, A. and Arsenin, V. (1977). Solutions of Ill-posed Problems, Scripta Series in Mathematics, John Wiley&Sons, Washington, DC.
  • Turner, R. (1986). A target field approach to optimal coil design, Journal of Physics D: Applied Physics 19(8): 147-151.
  • Voglis, C. and Lagaris, I. (2004). BOXCQP: An algorithm for bound constrained convex quadratic problems, Proceedings of the 1st International Conference: From Scientific Computing to Computational Engineering, IC-SCCE, Athens, Greece.
  • Xu, H., Conolly, S., Scott, G. and Macovski, A. (1999). Fundamental scaling relations for homogeneous magnets, ISMRM 7th Scientific Meeting, Philadelphia, PA, USA, p. 475.
  • Zhu, M., Xia, L., Liu, F., Zhu, J., L.Kang and Crozier, S. (2012). Finite difference method for the design of gradient coils in MRI-Initial framework, IEEE Transactions on Biomedical Engineering 59(9): 2412-2421.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv24i2p249bwm
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