Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-23T09:16:33.312Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  31 March 2010

Marcus Wagner
Marcus Wagner*
Affiliation:
University of Graz, Institute for Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria. www.thecitytocome.de; marcus.wagner@uni-graz.at
Get access

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 The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and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.

Keywords

Quasiconvex functions with infinite valueslower semicontinuous quasiconvex envelopemultidimensional control problemrelaxationexistence of global minimizersimage registrationpolyconvex regularization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 190 - 221
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1984) 125–145. CrossRef
Alvarez, L., Weickert, J. and Sánchez, J., Reliable estimation of dense optical flow fields with large displacements. Int. J. Computer Vision 39 (2000) 41–56. CrossRef
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).
Ball, J.M. and Murat, F., W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. CrossRef
N. Bourbaki, Éléments de Mathématique, Livre VI : Intégration, Chapitres I–IV. Hermann, Paris, France (1952).
Brokate, M., Pontryagin's principle for control problems in age-dependent population dynamics. J. Math. Biology 23 (1985) 75–101. CrossRef
A. Brøndsted, An Introduction to Convex Polytopes. Springer, New York-Heidelberg-Berlin (1983).
Brune, C., Maurer, H. and Wagner, M., Detection of intensity and motion edges within optical flow via multidimensional control. SIAM J. Imaging Sci. 2 (2009) 1190–1210. CrossRef
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics 207. Longman, Harlow (1989).
Conti, S., Quasiconvex functions incorporating volumetric constraints are rank-one convex. J. Math. Pures Appl. 90 (2008) 15–30. CrossRef
B. Dacorogna, Introduction to the Calculus of Variations. Imperial College Press, London, UK (2004)
B. Dacorogna, Direct Methods in the Calculus of Variations. Second edn., Springer, New York etc. (2008).
Dacorogna, B. and Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1–37. CrossRef
Droske, M. and Rumpf, M., A variational approach to nonrigid morphological image registration. SIAM J. Appl. Math. 64 (2004) 668–687. CrossRef
N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. Wiley-Interscience, New York etc. (1988).
I. Ekeland and R. Témam, Convex Analysis and Variational Problems. Second edn., SIAM, Philadelphia, USA (1999).
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton etc. (1992).
Feichtinger, G., Tragler, G. and Veliov, V.M., Optimality conditions for age-structured control systems. J. Math. Anal. Appl. 288 (2003) 47–68. CrossRef
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).
Gallardo, L.A. and Meju, M.A., Characterization of heterogeneous near-surface materials by joint 2D inversion of dc resistivity and seismic data. Geophys. Res. Lett. 30 (2003) 1658. CrossRef
Haber, E. and Modersitzki, J., Intensity gradient based registration and fusion of multi-modal images. Methods Inf. Med. 46 (2007) 292–299.
Henn, S. and Witsch, K., A multigrid approach for minimizing a nonlinear functional for digital image matching. Computing 64 (2000) 339–348. CrossRef
Henn, S. and Witsch, K., Iterative multigrid regularization techniques for image matching. SIAM J. Sci. Comput. 23 (2001) 1077–1093. CrossRef
Hermosillo, G., Chefd'hotel, C. and Faugeras, O., Variational methods for multimodal image matching. Int. J. Computer Vision 50 (2002) 329–343. CrossRef
Hinterberger, W., Scherzer, O., Schnörr, C. and Weickert, J., Analysis of optical flow models in the framework of the calculus of variations. Num. Funct. Anal. Optim. 23 (2002) 69–89. CrossRef
Kinderlehrer, D. and Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Rat. Mech. Anal. 115 (1991) 329–365. CrossRef
Marcellini, P. and Sbordone, C., Semicontinuity problems in the calculus of variations. Nonlinear Anal. 4 (1980) 241–257. CrossRef
J. Modersitzki, Numerical Methods for Image Registration. Oxford University Press, Oxford, UK (2004).
C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren 130. Springer, Berlin-Heidelberg-New York (1966).
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 Mathematics 411, Chapman & Hall/CRC Press, Boca Raton etc. (2000) 217–236.
T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. De Gruyter, Berlin-New York (1997).
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, UK (1993).
Ting, T.W., Elastic-plastic torsion of convex cylindrical bars. J. Math. Mech. 19 (1969) 531–551.
Ting, T.W., Elastic-plastic torsion problem III. Arch. Rat. Mech. Anal. 34 (1969) 228–244.
M. Wagner, Erweiterungen des mehrdimensionalen Pontrjaginschen Maximumprinzips auf meßbare und beschränkte sowie distributionelle Steuerungen. Ph.D. Thesis, University of Leipzig, Germany (1996).
M. Wagner, Mehrdimensionale Steuerungsprobleme mit quasikonvexen Integranden. Habilitation Thesis, BTU Cottbus, Germany (2006).
M. Wagner, Nonconvex relaxation properties of multidimensional control problems, in Recent Advances in Optimization, A. Seeger Ed., Lecture Notes in Economics and Mathematical Systems 563, Springer, Berlin etc. (2006) 233–250.
Wagner, M., Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl. 18 (2008) 305–327.
Wagner, M., Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems. J. Math. Anal. Appl. 355 (2009) 606–619. CrossRef
Wagner, M., On the lower semicontinuous quasiconvex envelope for unbounded integrands (I). ESAIM: COCV 15 (2009) 68–101. CrossRef
Wagner, M., On the lower semicontinuous quasiconvex envelope for unbounded integrands (II): Representation by generalized controls. J. Convex Anal. 16 (2009) 441–472.
Wagner, M., Pontryagin's maximum principle for multidimensional control problems in image processing. J. Optim. Theory Appl. 140 (2009) 543–576. CrossRef
M. Wagner, Elastic/hyperelastic image registration unter Nebenbedingungen als mehrdimensionales Steuerungsproblem. Preprint-Reihe Mathematik, Preprint Nr. M-09/2009, BTU Cottbus, Germany (2009).