Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)

Marcus Wagner

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 190-221
  • ISSN: 1292-8119

Abstract

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We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.

How to cite

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Wagner, Marcus. "Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 190-221. <http://eudml.org/doc/197289>.

@article{Wagner2011,
abstract = { We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms. },
author = {Wagner, Marcus},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Quasiconvex functions with infinite values; lower semicontinuous quasiconvex envelope; multidimensional control problem; relaxation; existence of global minimizers; image registration; polyconvex regularization; quasiconvex functions with infinite values},
language = {eng},
month = {2},
number = {1},
pages = {190-221},
publisher = {EDP Sciences},
title = {Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)},
url = {http://eudml.org/doc/197289},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Wagner, Marcus
TI - Quasiconvex relaxation of multidimensional control problems with integrands f(t, ξ, v)
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 190
EP - 221
AB - We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.
LA - eng
KW - Quasiconvex functions with infinite values; lower semicontinuous quasiconvex envelope; multidimensional control problem; relaxation; existence of global minimizers; image registration; polyconvex regularization; quasiconvex functions with infinite values
UR - http://eudml.org/doc/197289
ER -

References

top
  1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal.86 (1984) 125–145.  
  2. L. Alvarez, J. Weickert and J. Sánchez, Reliable estimation of dense optical flow fields with large displacements. Int. J. Computer Vision39 (2000) 41–56.  
  3. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Second edn., Springer, New York etc. (2006).  
  4. J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.  
  5. N. Bourbaki, Éléments de Mathématique, Livre VI : Intégration, Chapitres I–IV. Hermann, Paris, France (1952).  
  6. M. Brokate, Pontryagin's principle for control problems in age-dependent population dynamics. J. Math. Biology23 (1985) 75–101.  
  7. A. Brøndsted, An Introduction to Convex Polytopes. Springer, New York-Heidelberg-Berlin (1983).  
  8. C. Brune, H. Maurer and M. Wagner, Detection of intensity and motion edges within optical flow via multidimensional control. SIAM J. Imaging Sci.2 (2009) 1190–1210.  
  9. G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics207. Longman, Harlow (1989).  
  10. S. Conti, Quasiconvex functions incorporating volumetric constraints are rank-one convex. J. Math. Pures Appl.90 (2008) 15–30.  
  11. B. Dacorogna, Introduction to the Calculus of Variations. Imperial College Press, London, UK (2004)  
  12. B. Dacorogna, Direct Methods in the Calculus of Variations. Second edn., Springer, New York etc. (2008).  
  13. B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math.178 (1997) 1–37.  
  14. M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration. SIAM J. Appl. Math.64 (2004) 668–687.  
  15. N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. Wiley-Interscience, New York etc. (1988).  
  16. I. Ekeland and R. Témam, Convex Analysis and Variational Problems. Second edn., SIAM, Philadelphia, USA (1999).  
  17. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton etc. (1992).  
  18. G. Feichtinger, G. Tragler and V.M. Veliov, Optimality conditions for age-structured control systems. J. Math. Anal. Appl.288 (2003) 47–68.  
  19. L. Franek, M. Franek, H. Maurer and M. Wagner, Image restoration and simultaneous edge detection by optimal control methods. BTU Cottbus, Preprint-Reihe Mathematik, Preprint Nr. M-05/2008. Optim. Contr. Appl. Meth. (submitted).  
  20. L.A. Gallardo and M.A. Meju, Characterization of heterogeneous near-surface materials by joint 2D inversion of dc resistivity and seismic data. Geophys. Res. Lett.30 (2003) 1658.  
  21. E. Haber and J. Modersitzki, Intensity gradient based registration and fusion of multi-modal images. Methods Inf. Med.46 (2007) 292–299.  
  22. S. Henn and K. Witsch, A multigrid approach for minimizing a nonlinear functional for digital image matching. Computing64 (2000) 339–348.  
  23. S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching. SIAM J. Sci. Comput.23 (2001) 1077–1093.  
  24. G. Hermosillo, C. Chefd'hotel and O. Faugeras, Variational methods for multimodal image matching. Int. J. Computer Vision50 (2002) 329–343.  
  25. W. Hinterberger, O. Scherzer, C. Schnörr and J. Weickert, Analysis of optical flow models in the framework of the calculus of variations. Num. Funct. Anal. Optim.23 (2002) 69–89.  
  26. D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365.  
  27. P. Marcellini and C. Sbordone, Semicontinuity problems in the calculus of variations. Nonlinear Anal.4 (1980) 241–257.  
  28. J. Modersitzki, Numerical Methods for Image Registration. Oxford University Press, Oxford, UK (2004).  
  29. C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren130. Springer, Berlin-Heidelberg-New York (1966).  
  30. S. Pickenhain and M. Wagner, Critical points in relaxed deposit problems, in Calculus of Variations and Optimal Control, Technion 98, Vol. II, A. Ioffe, S. Reich and I. Shafrir Eds., Research Notes in Mathematics411, Chapman & Hall/CRC Press, Boca Raton etc. (2000) 217–236.  
  31. T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. De Gruyter, Berlin-New York (1997).  
  32. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, UK (1993).  
  33. T.W. Ting, Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech.19 (1969) 531–551.  
  34. T.W. Ting, Elastic-plastic torsion problem III. Arch. Rat. Mech. Anal.34 (1969) 228–244.  
  35. M. Wagner, Erweiterungen des mehrdimensionalen Pontrjaginschen Maximumprinzips auf meßbare und beschränkte sowie distributionelle Steuerungen. Ph.D. Thesis, University of Leipzig, Germany (1996).  
  36. M. Wagner, Mehrdimensionale Steuerungsprobleme mit quasikonvexen Integranden. Habilitation Thesis, BTU Cottbus, Germany (2006).  
  37. M. Wagner, Nonconvex relaxation properties of multidimensional control problems, in Recent Advances in Optimization, A. Seeger Ed., Lecture Notes in Economics and Mathematical Systems563, Springer, Berlin etc. (2006) 233–250.  
  38. M. Wagner, Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl.18 (2008) 305–327.  
  39. M. Wagner, Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems. J. Math. Anal. Appl.355 (2009) 606–619.  
  40. M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands (I). ESAIM: COCV15 (2009) 68–101.  
  41. M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands (II): Representation by generalized controls. J. Convex Anal.16 (2009) 441–472.  
  42. M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing. J. Optim. Theory Appl.140 (2009) 543–576.  
  43. M. Wagner, Elastic/hyperelastic image registration unter Nebenbedingungen als mehrdimensionales Steuerungsproblem. Preprint-Reihe Mathematik, Preprint Nr. M-09/2009, BTU Cottbus, Germany (2009).  

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