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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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.  Zbl0565.49010
  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.  Zbl1060.68635
  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).  Zbl1110.35001
  4. J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.  Zbl0549.46019
  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.  Zbl0599.92017
  7. A. Brøndsted, An Introduction to Convex Polytopes. Springer, New York-Heidelberg-Berlin (1983).  Zbl0509.52001
  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.  Zbl1181.49029
  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.  Zbl1146.49009
  11. B. Dacorogna, Introduction to the Calculus of Variations. Imperial College Press, London, UK (2004)  Zbl1095.49002
  12. B. Dacorogna, Direct Methods in the Calculus of Variations. Second edn., Springer, New York etc. (2008).  Zbl1140.49001
  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.  Zbl0901.49027
  14. M. Droske and M. Rumpf, A variational approach to nonrigid morphological image registration. SIAM J. Appl. Math.64 (2004) 668–687.  Zbl1063.49013
  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).  Zbl0939.49002
  17. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton etc. (1992).  Zbl0804.28001
  18. G. Feichtinger, G. Tragler and V.M. Veliov, Optimality conditions for age-structured control systems. J. Math. Anal. Appl.288 (2003) 47–68.  Zbl1042.49035
  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).  Zbl1301.49073
  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.  Zbl0961.65120
  23. S. Henn and K. Witsch, Iterative multigrid regularization techniques for image matching. SIAM J. Sci. Comput.23 (2001) 1077–1093.  Zbl0999.65057
  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.  Zbl1016.49002
  26. D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal.115 (1991) 329–365.  Zbl0754.49020
  27. P. Marcellini and C. Sbordone, Semicontinuity problems in the calculus of variations. Nonlinear Anal.4 (1980) 241–257.  Zbl0537.49002
  28. J. Modersitzki, Numerical Methods for Image Registration. Oxford University Press, Oxford, UK (2004).  Zbl1055.68140
  29. C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren130. Springer, Berlin-Heidelberg-New York (1966).  Zbl0142.38701
  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.  Zbl0960.49021
  31. T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. De Gruyter, Berlin-New York (1997).  Zbl0880.49002
  32. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, UK (1993).  Zbl0798.52001
  33. T.W. Ting, Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech.19 (1969) 531–551.  Zbl0197.23301
  34. T.W. Ting, Elastic-plastic torsion problem III. Arch. Rat. Mech. Anal.34 (1969) 228–244.  Zbl0179.53903
  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.  Zbl1108.49028
  38. M. Wagner, Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl.18 (2008) 305–327.  Zbl1154.49007
  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.  Zbl1162.49016
  40. M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands (I). ESAIM: COCV15 (2009) 68–101.  Zbl1173.26009
  41. M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands (II): Representation by generalized controls. J. Convex Anal.16 (2009) 441–472.  Zbl1186.26025
  42. M. Wagner, Pontryagin's maximum principle for multidimensional control problems in image processing. J. Optim. Theory Appl.140 (2009) 543–576.  Zbl1159.49033
  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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