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Convergence rates for Kaczmarz-type regularization methods
Geometric reconstruction in bioluminescence tomography
1. | Fakultät für Mathematik, Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie (KIT), D-76128 Karlsruhe, Germany, Germany |
References:
[1] |
K. Atkinson and W. Han, Theoretical Numerical Analysis, 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009.
doi: 10.1007/978-1-4419-0458-4. |
[2] |
H. Attouch, G. Buttazzo and G.Michaille, Variational Analysis in Sobolev and BV Space, MPS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. |
[3] |
G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp.
doi: 10.1088/0266-5611/25/5/053001. |
[4] |
M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988. |
[5] |
M. Burger and S. J. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301.
doi: 10.1017/S0956792505006182. |
[6] |
F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157.
doi: 10.3934/ipi.2013.7.123. |
[7] |
W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner and A. Cong, Practical reconstruction method for bioluminescence tomography, Opt. Express, 13 (2005), 6756-6771.
doi: 10.1364/OPEX.13.006756. |
[8] |
C. H. Contag and B. D. Ross, It's not just about anatomy: In vivo bioluminescence imaging as an eyepiece into biology, Journal of Magnetic Resonance Imaging, 16 (2002), 378-387.
doi: 10.1002/jmri.10178. |
[9] |
A. De Cezaro and A. Leitão, Level-set approaches of $L_2$-type for recovering shape and contrast in ill-posed problems, Inverse Probl. Sci. Eng., 20 (2012), 571-587.
doi: 10.1080/17415977.2011.639452. |
[10] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
doi: 10.1137/1.9780898719826. |
[11] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[12] |
______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474.
doi: 10.1090/S0273-0979-1979-14595-6. |
[13] |
W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1998. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser, Boston, 1984. |
[15] |
W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675.
doi: 10.1088/0266-5611/22/5/008. |
[16] |
M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens, 3rd edition, Vieweg + Teubner, Wiesbaden, 2009.
doi: 10.1007/978-3-8348-9309-3. |
[17] |
H. Harbrecht and J. Tausch, An efficient numerical method for a shape-identification problem arising from the heat equation, Inverse Problems, 27 (2011), 065013, 18 pp.
doi: 10.1088/0266-5611/27/6/065013. |
[18] |
F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, Springer, 2009.
doi: 10.1007/978-1-4020-8839-1_3. |
[20] |
C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999.
doi: 10.1137/1.9781611970920. |
[21] |
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006. |
[22] |
R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557.
doi: 10.1016/j.jcp.2006.06.041. |
[23] |
_______, Regularization of ill-posed Mumford-Shah models with perimeter penalization, Inverse Problems, 26 (2010), 115001, 25 pp.
doi: 10.1088/0266-5611/26/11/115001. |
[24] |
J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687.
doi: 10.1080/01630563.1980.10120631. |
[25] |
G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299.
doi: 10.1118/1.1766420. |
[26] |
R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128.
doi: 10.1038/nm0103-123. |
[27] |
H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472.
doi: 10.2307/2371513. |
show all references
References:
[1] |
K. Atkinson and W. Han, Theoretical Numerical Analysis, 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009.
doi: 10.1007/978-1-4419-0458-4. |
[2] |
H. Attouch, G. Buttazzo and G.Michaille, Variational Analysis in Sobolev and BV Space, MPS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006. |
[3] |
G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48 pp.
doi: 10.1088/0266-5611/25/5/053001. |
[4] |
M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces, Graduate Texts in Mathematics, 115, Springer, New York, 1988. |
[5] |
M. Burger and S. J. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), 263-301.
doi: 10.1017/S0956792505006182. |
[6] |
F. Caubet, M. Dambrine, D. Kateb and C. Z. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Probl. Imaging, 7 (2013), 123-157.
doi: 10.3934/ipi.2013.7.123. |
[7] |
W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner and A. Cong, Practical reconstruction method for bioluminescence tomography, Opt. Express, 13 (2005), 6756-6771.
doi: 10.1364/OPEX.13.006756. |
[8] |
C. H. Contag and B. D. Ross, It's not just about anatomy: In vivo bioluminescence imaging as an eyepiece into biology, Journal of Magnetic Resonance Imaging, 16 (2002), 378-387.
doi: 10.1002/jmri.10178. |
[9] |
A. De Cezaro and A. Leitão, Level-set approaches of $L_2$-type for recovering shape and contrast in ill-posed problems, Inverse Probl. Sci. Eng., 20 (2012), 571-587.
doi: 10.1080/17415977.2011.639452. |
[10] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
doi: 10.1137/1.9780898719826. |
[11] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[12] |
______, Nonconvex minization problems, Bull. Am. Math. Soc., New Ser., 1 (1979), 443-474.
doi: 10.1090/S0273-0979-1979-14595-6. |
[13] |
W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 1998. |
[14] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser, Boston, 1984. |
[15] |
W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675.
doi: 10.1088/0266-5611/22/5/008. |
[16] |
M. Hanke-Bourgeois, Grundlagen der Numerischen Mathematik und des Wissenschaftlichen Rechnens, 3rd edition, Vieweg + Teubner, Wiesbaden, 2009.
doi: 10.1007/978-3-8348-9309-3. |
[17] |
H. Harbrecht and J. Tausch, An efficient numerical method for a shape-identification problem arising from the heat equation, Inverse Problems, 27 (2011), 065013, 18 pp.
doi: 10.1088/0266-5611/27/6/065013. |
[18] |
F. Hettlich, The Domain Derivative in Inverse Obstacle Problems, Habilitation thesis, Friedrich-Alexander-Universität, Erlangen, 1999. |
[19] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, Springer, 2009.
doi: 10.1007/978-1-4020-8839-1_3. |
[20] |
C. T. Kelley, Iterative Methods for Optimization, Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999.
doi: 10.1137/1.9781611970920. |
[21] |
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operation Research and Financial Engineering, Springer, New York, 2006. |
[22] |
R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data, J. Comput. Phys., 221 (2007), 539-557.
doi: 10.1016/j.jcp.2006.06.041. |
[23] |
_______, Regularization of ill-posed Mumford-Shah models with perimeter penalization, Inverse Problems, 26 (2010), 115001, 25 pp.
doi: 10.1088/0266-5611/26/11/115001. |
[24] |
J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., 2 (1980), 649-687.
doi: 10.1080/01630563.1980.10120631. |
[25] |
G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography, Medical Physics 31 (2004), 2289-2299.
doi: 10.1118/1.1766420. |
[26] |
R. Weissleder and V. Ntziachristos, Shedding light onto live molecular targets, Nat. Med., 9 (2003), 123-128.
doi: 10.1038/nm0103-123. |
[27] |
H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472.
doi: 10.2307/2371513. |
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