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Erschienen in:
Linear Inverse Problems Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

Iterative Solution Methods

verfasst von : Martin Burger, Barbara Kaltenbacher, Andreas Neubauer

Erschienen in: Handbook of Mathematical Methods in Imaging

Verlag: Springer New York

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Abstract

This chapter deals with iterative methods for nonlinear ill-posed problems. We present gradient and Newton type methods as well as nonstandard iterative algorithms such as Kaczmarz, expectation maximization, and Bregman iterations. Our intention here is to cite convergence results in the sense of regularization and to provide further references to the literature.

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Vorheriges Kapitel EM Algorithms from a Non-stochastic Perspective
Nächstes Kapitel Level Set Methods for Structural Inversion and Image Reconstruction
Literatur
1.
Akcelik, V., Biros, G., Draganescu, A., Hill, J., Ghattas, O., Waanders, B.V.B.: Dynamic data-driven inversion for terascale simulations: real-time identification of airborne contaminants. In: Proceedings of SC05. IEEE/ACM, Seattle (2005)
2.
Bachmayr, M., Burger, M.: Iterative total variation schemes for nonlinear inverse problems. Inverse Prob. 25, 105,004 (2009)CrossRefMathSciNet
3.
Bakushinsky, A.B.: The problem of the convergence of the iteratively regularized Gauss-Newton method. Comput. Math. Math. Phys. 32, 1353–1359 (1992)MathSciNet
4.
Bakushinsky, A.B.: Iterative methods without degeneration for solving degenerate nonlinear operator equations. Dokl. Akad. Nauk. 344, 7–8 (1995)MathSciNet
5.
Bakushinsky, A.B., Kokurin, M.Y.: Iterative Methods for Approximate Solution of Inverse Problems. Mathematics and Its Applications, vol. 577. Springer, Dordrecht (2004)
6.
7.
Bauer, F., Kindermann, S.: The quasi-optimality criterion for classical inverse problems. Inverse Prob. 24(3), 035,002 (20 pp) (2008)
8.
Bauer, F., Hohage, T., Munk, A.: Iteratively regularized Gauss-Newton method for nonlinear inverse problems with random noise. SIAM J. Numer. Anal. 47, 1827–1846 (2009)CrossRefMATHMathSciNet
9.
Baumeister, J., Cezaro, A.D., Leitao, A.: On iterated Tikhonov-Kaczmarz regularization methods for ill-posed problems. ICJV (2010). doi:10.1007/s11263-010-0339-5
10.
Baumeister, J., Kaltenbacher, B., Leitao, A.: On Levenberg-Marquardt Kaczmarz regularization methods for ill-posed problems. Inverse Prob. Imaging 4, 335–350 (2010)CrossRefMATHMathSciNet
11.
Benning, M., Brune, C., Burger, M., Mueller, J.: Higher-order tv methods - enhancement via bregman iteration. J. Sci. Comput. 54(2-3), 269–310 (2013). In honor of Stanley Osher for his 70th birthdayCrossRefMATHMathSciNet
12.
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. Institute of Physics Publishing, Bristol (1998)CrossRefMATH
13.
Bissantz, N., Hohage, T., Munk, A., Ruymgaart, F.: Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45, 2610–2636 (2007)CrossRefMATHMathSciNet
14.
Bissantz, N., Mair, B., Munk, A.: A statistical stopping rule for MLEM reconstructions in PET. IEEE Nucl. Sci. Symp. Conf. Rec 8, 4198–4200 (2008)
15.
Blaschke, B., Neubauer, A., Scherzer, O.: On convergence rates for the iteratively regularized Gauss-Newton method. IMA J. Numer. Anal. 17, 421–436 (1997)CrossRefMATHMathSciNet
16.
Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp. Math. Math. Phys. 7, 200–217 (1967)CrossRef
17.
Brune, C., Sawatzky, A., Burger, M.: Bregman-EM-TV methods with application to optical nanoscopy. In: Tai, X.-C., et al. (ed.) Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. Lecturer Notes in Computer Science, vol. 5567, pp. 235–246. Springer, New York (2009)CrossRef
18.
Brune, C., Sawatzky, A., Burger, M.: Primal and dual Bregman methods with application to optical nanoscopy. Int. J. Comput. Vis 92, 211–229 (2011)CrossRefMATHMathSciNet
19.
Burger, M., Kaltenbacher, B.: Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems. SIAM J. Numer. Anal. 44, 153–182 (2006)CrossRefMATHMathSciNet
20.
Burger, M., Resmerita, E., He, L.: Error estimation for Bregman iterations and inverse scale space methods in image restoration. Computing 81(2-3), 109–135 (2007)CrossRefMATHMathSciNet
21.
Cai, J.F., Osher, S., Shen, Z.: Convergence of the linearized Bregman iteration for l1-norm minimization. Math. Comput. 78, 2127–2136 (2009)CrossRefMATHMathSciNet
22.
23.
Cai, J.F., Osher, S., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2, 226–252 (2009)CrossRefMATHMathSciNet
24.
25.
Combettes, P.L., Pesquet, J.C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Prob. 24, 065,014 (27 pp) (2008)
26.
Daubechies, I., Defrise, M., Mol, C.D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413–1457 (2004)CrossRefMATHMathSciNet
27.
De Cezaro, A., Haltmeier, M., Leitao, A., Scherzer, O.: On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations. Appl. Math. Comput. 202, 596–607 (2008)CrossRefMATHMathSciNet
28.
29.
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum Likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38 (1977)MATHMathSciNet
30.
Deuflhard, P., Engl, H.W., Scherzer, O.: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Prob. 14, 1081–1106 (1998)CrossRefMATHMathSciNet
31.
Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)CrossRefMATHMathSciNet
32.
33.
34.
35.
Eicke, B., Louis, A.K., Plato, R.: The instability of some gradient methods for ill-posed problems. Numer. Math. 58, 129–134 (1990)CrossRefMATHMathSciNet
36.
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefMATH
37.
Engl, H.W., Zou, J.: A new approach to convergence rate analysis of tiknonov regularization for parameter identification in heat conduction. Inverse Prob. 16, 1907–1923 (2000)CrossRefMATHMathSciNet
38.
Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)CrossRefMATH
39.
Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imaging 9, 84–93 (1990)CrossRef
40.
Green, P.J.: On use of the EM algorithm for penalized likelihood estimation. J. R. Stat. Soc. Ser. B (Methodological) 52, 443–452 (1990)MATH
41.
42.
Haber, E., Ascher, U.: A multigrid method for distributed parameter estimation problems. Inverse Prob. 17, 1847–1864 (2001)CrossRefMATHMathSciNet
43.
Haltmeier, M., Leitao, A., Scherzer, O.: Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Prob. Imaging 1, 289–298 (2007)CrossRefMATHMathSciNet
44.
Haltmeier, M., Kowar, R., Leitao, A., Scherzer, O.: Kaczmarz methods for regularizing nonlinear ill-posed equations II: applications. Inverse Prob. Imaging 1, 507–523 (2007)CrossRefMATHMathSciNet
45.
Haltmeier, M., Leitao, A., Resmerita, E.: On regularization methods of EM-Kaczmarz type. Inverse Prob. 25(7), 17 (2009)CrossRefMathSciNet
46.
Hanke, M.: A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Prob. 13, 79–95 (1997)CrossRefMATHMathSciNet
47.
Hanke, M.: Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numer. Funct. Anal. Optim. 18, 971–993 (1997)CrossRefMATHMathSciNet
48.
49.
Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)CrossRefMATHMathSciNet
50.
He, L., Burger, M., Osher, S.: Iterative total variation regularization with non-quadratic fidelity. J. Math. Imaging Vis. 26, 167–184 (2006)CrossRefMathSciNet
51.
Hein, T., Kazimierski, K.: Modified Landweber iteration in Banach spaces – convergence and convergence rates. Numer. Funct. Anal. Optim. 31(10), 1158–1184 (2010)CrossRefMATHMathSciNet
52.
Hochbruck, M., Hönig, M., Ostermann, A.: A convergence analysis of the exponential euler iteration for nonlinear ill-posed problems. Inverse Prob. 25, 075,009 (18 pp) (2009)
53.
Hochbruck, M., Hönig, M., Ostermann, A.: Regularization of nonlinear ill-posed problems by exponential integrators. Math. Mod. Numer. Anal. 43, 709–720 (2009)CrossRefMATH
54.
Hohage, T.: Logarithmic convergence rates of the iteratively regularized Gauß-Newton method for an inverse potential and an inverse scattering problem. Inverse Prob. 13, 1279–1299 (1997)CrossRefMATHMathSciNet
55.
Hohage, T.: Iterative methods in inverse obstacle scattering: regularization theory of linear and nonlinear exponentially ill-posed problems. Ph.D. thesis, University of Linz (1999)
56.
57.
Hohage, T., Werner, F.: Iteratively regularized Newton methods with general data misfit functionals and applications to Poisson data. Numer. Math. 123, 745–779 (2013)CrossRefMATHMathSciNet
58.
Jin, Q.: Inexact Newton-Landweber iteration for solving nonlinear inverse problems in Banach spaces. Inverse Prob. 28(6), 15 (2012)CrossRef
59.
Kaltenbacher, B.: Some Newton type methods for the regularization of nonlinear ill-posed problems. Inverse Prob. 13, 729–753 (1997)CrossRefMATHMathSciNet
60.
61.
Kaltenbacher, B.: A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numer. Math. 79, 501–528 (1998)CrossRefMATHMathSciNet
62.
Kaltenbacher, B.: A projection-regularized Newton method for nonlinear ill-posed problems and its application to parameter identification problems with finite element discretization. SIAM J. Numer. Anal. 37, 1885–1908 (2000)CrossRefMATHMathSciNet
63.
Kaltenbacher, B.: On the regularizing properties of a full multigrid method for ill-posed problems. Inverse Prob. 17, 767–788 (2001)CrossRefMATHMathSciNet
64.
Kaltenbacher, B.: Convergence rates for the iteratively regularized Landweber iteration in Banach space. In: Hmberg, D., Trltzsch, F. (eds.) System Modeling and Optimization. 25th IFIP TC 7 Conference on System Modeling and Optimization, CSMO 2011, Berlin, Germany, September 12–16, 2011. Revised Selected Papers, pp. 38–48. Springer, Heidelberg (2013)
65.
Kaltenbacher, B., Hofmann, B.: Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Prob. 26, 035,007 (2010)CrossRefMathSciNet
66.
Kaltenbacher, B., Neubauer, A.: Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions. Inverse Prob. 22, 1105–1119 (2006)CrossRefMATHMathSciNet
67.
Kaltenbacher, B., Schicho, J.: A multi-grid method with a priori and a posteriori level choice for the regularization of nonlinear ill-posed problems. Numer. Math. 93, 77–107 (2002)MATHMathSciNet
68.
Kaltenbacher, B., Tomba, I.: Convergence rates for an iteratively regularized Newton-Landweber iteration in Banach space. Inverse Prob. 29, 025010 (2013). doi:10.1088/0266-5611/29/2/025010CrossRefMathSciNet
69.
Kaltenbacher, B., Neubauer, A., Ramm, A.G.: Convergence rates of the continuous regularized Gauss-Newton method. J. Inverse Ill-Posed Prob. 10, 261–280 (2002)MATHMathSciNet
70.
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics. de Gruyter, Berlin (2008)CrossRefMATH
71.
Kaltenbacher, B., Schöpfer, F., Schuster, T.: Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems. Inverse Prob. 25, 065,003 (19 pp) (2009)
72.
Kaltenbacher, B., Kirchner, A., Veljovic, S.: Goal oriented adaptivity in the IRGNM for parameter identification in PDEs I: reduced formulation Inverse Prob. 30, 045001 (2014)
73.
Kaltenbacher, B., Kirchner, A., Vexler, B.: Goal oriented adaptivity in the IRGNM for parameter identification in PDEs II: all-at once formulations Inverse Prob. 30, 045002 (2014)
74.
Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. ETNA 38, 233–257 (2011)MATHMathSciNet
75.
Kindermann, S., Neubauer, A.: On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Prob. Imaging 2, 291–299 (2008)CrossRefMATHMathSciNet
76.
77.
Kowar, R., Scherzer, O.: Convergence analysis of a Landweber-Kaczmarz method for solving nonlinear ill-posed problems. In: Romanov, V.G., Kabanikhin, S.I., Anikonov, Y.E., Bukhgeim, A.L. (eds.) Ill-Posed and Inverse Problems, pp. 69–90. VSP, Zeist (2002)
78.
79.
Kügler, P.: A derivative free Landweber iteration for parameter identification in certain elliptic PDEs. Inverse Prob. 19, 1407–1426 (2003)CrossRefMATH
80.
Kügler, P.: A derivative free Landweber method for parameter identification in elliptic partial differential equations with application to the manufacture of car windshields. Ph.D. thesis, Johannes Kepler University, Linz, Austria (2003)
81.
Langer, S.: Preconditioned Newton methods for ill-posed problems. Ph.D. thesis, University of Göttingen (2007)
82.
Langer, S., Hohage, T.: Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions. J. Inverse Ill-Posed Prob. 15, 19–35 (2007)CrossRefMathSciNet
83.
Leitao, A., Marques Alves, M.: On Landweber-Kaczmarz methods for regularizing systems of ill-posed equations in Banach spaces. Inverse Prob. 28, 104,008 (15 pp) (2012)
84.
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)CrossRefMATHMathSciNet
85.
Mülthei, H.N., Schorr, B.: On properties of the iterative maximum likelihood reconstruction method. Math. Methods Appl. Sci. 11, 331–342 (1989)CrossRefMATHMathSciNet
86.
Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
87.
88.
89.
Neubauer, A.: The convergence of a new heuristic parameter selection criterion for general regularization methods. Inverse Prob. 24, 055,005 (10 pp) (2008)
90.
Neubauer, A., Scherzer, O.: A convergent rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems. ZAA 14, 369–377 (1995)MATHMathSciNet
91.
Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation based image restoration. SIAM Multiscale Model. Simul. 4, 460–489 (2005)CrossRefMATHMathSciNet
92.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)
93.
Resmerita, E., Engl, H.W., Iusem, A.N.: The expectation-maximization algorithm for ill-posed integral equations: a convergence analysis. Inverse Prob. 23, 2575–2588 (2007)CrossRefMATHMathSciNet
94.
Rieder, A.: On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Prob. 15, 309–327 (1999)CrossRefMATHMathSciNet
95.
96.
Rieder, A.: Inexact Newton regularization using conjugate gradients as inner iteration. SIAM J. Numer. Anal. 43, 604–622 (2005)CrossRefMATHMathSciNet
97.
Sawatzky, A., Brune, C., Wübbeling, F., Kösters, T., Schäfers, K., Burger, M.: Accurate EM-TV algorithm in PET with low SNR. Nuclear Science Symposium Conference Record, 2008. NSS ’08, pp. 5133–5137. IEEE, New York (2008)
98.
Scherzer, O.: A modified Landweber iteration for solving parameter estimation problems. Appl. Math. Optim. 38, 45–68 (1998)CrossRefMATHMathSciNet
99.
Schöpfer, F., Louis, A.K., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Prob. 22, 311–329 (2006)CrossRefMATH
100.
Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Prob. 24, 055,008 (20pp) (2008)
101.
Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics. de Gruyter, Berlin/New York (2012)CrossRefMATH
102.
Stück, R., Burger, M., Hohage, T.: The iteratively regularized gauss-newton method with convex constraints and applications in 4pi-microscopy. Inverse Prob. 28, 015,012 (2012)CrossRef
103.
Vardi, Y., Shepp, L.A., Kaufman, L.: A statistical model for positron emission tomography with discussion. J. Am. Stat. Assoc. 80, 8–37 (1985)CrossRefMATHMathSciNet
Metadaten
Titel
Iterative Solution Methods
verfasst von
Martin Burger
Barbara Kaltenbacher
Andreas Neubauer
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Handbook of Mathematical Methods in Imaging

Print ISBN: 978-1-4939-0789-2

Electronic ISBN: 978-1-4939-0790-8

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-0790-8_9