Abstract
In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.
Keywords
- Hilbert Space
- Variational Inequality
- Regularization Parameter
- Tikhonov Regularization
- Neighboring Problem
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References
Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)
Ammari, H.: An Introduction to Mathematics of Emerging Biomedical Imaging. Springer, Berlin (2008)
Anzengruber, S.W., Hofmann, B., Mathé, P.: Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces. Appl. Anal. (2013). http://dx.doi.org/10.1080/00036811.2013.833326.
Anzengruber, S.W., Hofmann, B., Ramlau, R.: On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization. Inverse Probl. 29(12), 125002(21pp) (2013)
Anzengruber, S.W., Ramlau, R.: Morozov’s discrepancy principle for Tikhonov-type functionals with nonlinear operators. Inverse Probl. 26(2), 025001(17pp) (2010)
Anzengruber, S.W., Ramlau, R.: Convergence rates for Morozov’s discrepancy principle using variational inequalities. Inverse Probl. 27(10), 105007(18pp) (2011)
Bakushinsky, A., Goncharsky, A.: Ill-Posed Problems: Theory and Applications. Kluwer, Dordrecht (1994)
Bakushinsky, A.B., Kokurin, M.Yu.: Iterative Methods for Approximate Solution of Inverse Problems. Springer, Dordrecht (2004)
Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Birkhäuser, Boston, (1989)
Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig (1987)
Baumeister, J.: Deconvolution of appearance potential spectra. In: Kleinman R., Kress R., Martensen E. (eds.) Direct and Inverse Boundary Value Problems. Methoden und Verfahren der mathematischen Physik, vol. 37, pp. 1–13. Peter Lang, Frankfurt am Main (1991)
Benning, M., Burger, M.: Error estimates for general fidelities. Electron. Trans. Numer. Anal. 38, 44–68 (2011)
Benning, M., Burger, M.: Ground states and singular vectors of convex variational regularization methods. arXiv:1211.2057v1 (2012)
Beretta, E., Vessella, S.: Stable determination of boundaries from Cauchy data. SIAM J. Math. Anal. 30, 220–232 (1999)
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. Institute of Physics Publishing, Bristol (1998)
Bonesky, T., Kazimierski, K., Maass, P., Schöpfer, F., Schuster, T.: Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. Art. ID 192679, 19pp (2008)
Boţ, R.I., Hofmann, B.: An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems. J. Integral Equ. Appl. 22(3), 369–392 (2010)
Boţ, R.I., Hofmann, B.: The impact of a curious type of smoothness conditions on convergence rates in \(\boldsymbol{\ell^{1}}\)-regularization. Eurasian J. Math. Comput. Appl. 1(1), 29–40 (2013)
Bredies, K., Lorenz, D.A.: Regularization with non-convex separable constraints. Inverse Probl. 25(8), 085011(14pp) (2009)
Bukhgeim, A.L., Cheng, J., Yamamoto, M.: Stability for an inverse boundary problem of determining a part of a boundary. Inverse Probl 15, 1021–1032 (1999)
Burger, M., Flemming, J., Hofmann, B.: Convergence rates in ℓ 1-regularization if the sparsity assumption fails. Inverse Probl. 29(2), 025013(16pp) (2013)
Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20(5), 1411–1421 (2004)
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)
Bürger, S., Hofmann, B.: About a deficit in low order convergence rates on the example of autoconvolution. Appl. Anal. (2014, to appear). Preprint 2013–17, Preprintreihe der Fakultät für Mathematik der TU Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-130630.
Chavent, G.: Nonlinear Least Squares for Inverse Problems. Springer, Dordrecht (2009)
Chavent, G., Kunisch, K.: On weakly nonlinear inverse problems. SIAM J. Appl. Math. 56(2), 542–572 (1996)
Chavent, G., Kunisch, K.: State space regularization: geometric theory. Appl. Math. Opt. 37(3), 243–267 (1998)
Cheng, J., Hofmann, B., Lu, S.: The index function and Tikhonov regularization for ill-posed problems. J. Comput. Appl. Math. (2013). http://dx.doi.org/10.1016/j.cam.2013.09.035.
Cheng, J., Nakamura, G.: Stability for the inverse potential problem by finite measurements on the boundary. Inverse Probl. 17, 273–280 (2001)
Cheng, J., Yamamoto, M.: Conditional stabilizing estimation for an integral equation of first kind with analytic kernel. J. Integral Equ. Appl. 12, 39–61 (2000)
Cheng, J., Yamamoto, M.: One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Probl. 16(4), L31–L38 (2000)
Clason, C.: L ∞ fitting for inverse problems with uniform noise. Inverse Probl. 28(10), 104007(18pp) (2012)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, New York (2013)
Dai, Z., Lamm, P.K.: Local regularization for the nonlinear inverse autoconvolution problem. SIAM J. Numer. Anal. 46(2), 832–868 (2008)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996, 2000)
Engl, H.W., Isakov, V.: On the identifiability of steel reinforcement bars in concrete from magnetostatic measurements. Eur. J. Appl. Math. 3, 255–262 (1992)
Engl, H.W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularisation of non-linear ill-posed problems. Inverse Probl. 5(4), 523–540 (1989)
Engl, H.W., Zou, J.: A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction. Inverse Probl. 16(6), 1907–1923 (2000)
Favaro, P., Soatto, S. 3-D Shape Estimation and Image Restoration. Exploiting Defocus and Motion Blur. Springer, London (2007)
Fischer, B., Modersitzki, J.: Ill-posed medicine – an introduction to image registration. Inverse Probl. 24(3), 034008(16pp) (2008)
Flemming, J.: Generalized Tikhonov Regularization and Modern Convergence Rate Theory in Banach Spaces. Shaker Verlag, Aachen (2012)
Flemming, J.: Variational smoothness assumptions in convergence rate theory – an overview. J. Inverse Ill-Posed Probl. 21(3), 395–409 (2013)
Flemming, J.: Regularization of autoconvolution and other ill-posed quadratic equations by decomposition. J. Inverse and Ill-Posed Probl. 22 (2014). doi:10.1515/jip-2013-0038
Flemming, J., Hofmann, B.: A new approach to source conditions in regularization with general residual term. Numer. Funct. Anal. Optim. 31(3), 254–284 (2010)
Flemming, J., Hofmann, B.: Convergence rates in constrained Tikhonov regularization: equivalence of projected source conditions and variational inequalities. Inverse Probl. 27(8), 085001(11pp) (2011)
Fleischer, G., Hofmann, B.: On inversion rates for the autoconvolution equation. Inverse Probl. 12(4), 419–435 (1996)
Gerth, D., Hofmann, B., Birkholz, S., Koke, S., Steinmeyer, G.: Regularization of an autoconvolution problem in ultrashort laser pulse characterization. Inverse Probl. Sci. Eng. 22(2), 245–266 (2014)
Gorenflo, R., Hofmann, B.: On autoconvolution and regularization. Inverse Probl. 10(2), 353–373 (1994)
Grasmair, M.: Well-posedness and convergence rates for sparse regularization with sublinear l q penalty term. Inverse Probl. Imaging 3(3), 383–387 (2009)
Grasmair, M.: Generalized Bregman distances and convergence rates for non-convex regularization methods. Inverse Probl. 26(11), 115014(16pp) (2010)
Grasmair, M.: Variational inequalities and higher order convergence rates for Tikhonov regularisation on Banach spaces. J. Inverse Ill-Posed Probl. 21(3), 379–394 (2013)
Grasmair, M., Haltmeier, M., Scherzer, O.: Sparse regularization with ℓ q penalty term. Inverse Probl. 24(5), 055020(13pp) (2008)
Grasmair, M., Haltmeier, M., Scherzer, O.: The residual method for regularizing ill-posed problems. Appl. Math. Comput. 218(6), 2693–2710 (2011)
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. Pitman, Boston (1984)
Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)
Hao, D.N., Quyen, T.N.T.: Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations. Inverse Probl. 26(12), 125014(23pp) (2010)
Hein, T.: Tikhonov regularization in Banach spaces – improved convergence rates results. Inverse Probl. 25(3), 035002(18pp) (2009)
Hein, T., Hofmann, B.: Approximate source conditions for nonlinear ill-posed problems – chances and limitations. Inverse Probl. 25(3), 035003(16pp) (2009)
Hofmann, B.: Regularization for Applied Inverse and Ill-Posed Problems. Teubner, Leipzig (1986)
Hofmann, B., Kaltenbacher, B., Pöschl, C., Scherzer, O.: A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Probl. 23(3), 987–1010 (2007)
Hofmann, B., Mathé, P.: Analysis of profile functions for general linear regularization methods. SIAM J. Numer. Anal. 45(3), 1122–1141 (2007)
Hofmann, B., Mathé, P.: Parameter choice in Banach space regularization under variational inequalities. Inverse Probl. 28(10), 104006(17pp) (2012)
Hofmann, B., Mathé, P., Pereverzev, S.V.: Regularization by projection: approximation theoretic aspects and distance functions. J. Inverse Ill-Posed Probl. 15(5), 527–545 (2007)
Hofmann, B., Scherzer, O.: Factors influencing the ill-posedness of nonlinear problems. Inverse Probl. 10(6), 1277–1297 (1994)
Hofmann, B., Yamamoto, M.: On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems. Appl. Anal. 89(11), 1705–1727 (2010)
Hohage, T., Pricop, M.: Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Probl. Imaging 2(2), 271–290 (2008)
Imanuvilov, O.Yu., Yamamoto, M.: Global uniqueness and stability in determining coefficients of wave equations. Commun. Partial Differ. Equ. 26, 1409–1425 (2001)
Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)
Ito, K., Kunisch, K.: On the choice of the regularization parameter in nonlinear inverse problems. SIAM J. Optim. 2(3), 376–404 (1992)
Janno, J., Wolfersdorf, L.v.: A general class of autoconvolution equations of the third kind. Z. Anal. Anwendungen 24(3), 523–543 (2005)
Jiang, D., Feng, H., Zou, J.: Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system. Inverse Probl. 28(10), 104002(20pp) (2012)
Jin, B., Zou, J.: Augmented Tikhonov regularization. Inverse Probl. 25(2), 025001(25pp) (2009)
Kabanikhin, S.I.: Inverse and Ill-Posed Problems – Theory and Applications. Inverse and Ill-Posed Problems Series, vol. 55. Walter de Gruyter, Berlin (2011)
Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, New York (2005)
Kaltenbacher, B.: A note on logarithmic convergence rates for nonlinear Tikhonov regularization. J. Inverse Ill-Posed Probl. 16(1), 79–88 (2008)
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Walter de Gruyter, Berlin (2008)
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. Springer, New York (2011)
Klann, E., Kuhn, M., Lorenz, D.A., Maass, P., Thiele, H.: Shrinkage versus deconvolution. Inverse Probl. 23(5), 2231–2248 (2007)
Kress, R.: Linear Integral Equations, 2nd edn. Springer, New York (1999)
Lamm, P.K., Dai, Z.: On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type. Inverse Probl. 21(5), 1773–1790 (2005)
Lattès, R., Lions, J.-L.: The Method of Quasi-Reversibility. Applications to Partial Differential Equations. Modern Analytic and Computational Methods in Science and Mathematics, vol. 18. American Elsevier, New York (1969)
Lorenz, D., Rösch, A.: Error estimates for joint Tikhonov and Lavrentiev regularization of constrained control problems. Appl. Anal. 89(11), 1679–1691 (2010)
Lorenz, D., Worliczek, N.: Necessary conditions for variational regularization schemes. Inverse Probl. 29(7), 075016(19pp), (2013)
Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart (1989)
Louis, A.K.: Approximate inverse for linear and some nonlinear problems. Inverse Probl. 11(6), 1211–1223 (1995)
Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6(4), 313–344 (1998)
Lu, S., Flemming, J.: Convergence rate analysis of Tikhonov regularization for nonlinear ill-posed problems with noisy operators. Inverse Probl. 28(10), 104003(19pp) (2012)
Lu, S., Pereverzev, S.V., Ramlau, R.: An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption. Inverse Probl. 23(1), 217–230 (2007)
Mahale, P., Nair, M.T.: Tikhonov regularization of nonlinear ill-posed equations under general source conditions. J. Inverse Ill-Posed Probl. 15(8), 813–829 (2007)
Mathé, P., Pereverzev, S.V.: Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Probl. 19(3), 789–803 (2003)
Modersitzki, J.: FAIR. Flexible Algorithms for Image Registration. SIAM, Philadelphia (2009)
Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)
Natterer, F.: Imaging and inverse problems of partial differential equations. Jahresber. Dtsch. Math.-Ver. 109(1), 31–48 (2007)
Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)
Neubauer, A.: Tikhonov regularization for nonlinear ill-posed problems: optimal convergence rate and finite dimensional approximation. Inverse Probl. 5(4), 541–558 (1989)
Neubauer, A.: On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. Inverse Probl. 25(6), 065009(10pp) (2009)
Neubauer, A., Hein, T., Hofmann, B., Kindermann, S., Tautenhahn, U.: Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces. Appl. Anal. 89(11), 1729–1743 (2010)
Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. ACM 9(1), 84–97 (1962)
Pöschl, C.: Tikhonov Regularization with General Residual Term. PhD thesis, University of Innsbruck, Austria, (2008)
Ramlau, R.: Morozov’s discrepancy principle for Tikhonov-regularization of nonlinear operators. Numer. Funct. Anal. and Optim. 23(1–2), 147–172 (2002)
Ramlau, R.: TIGRA – an iterative algorithm for regularizing nonlinear ill-posed problems. Inverse Probl. 19(2), 433–465 (2003)
Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21(4), 1303–1314 (2005)
Resmerita, E., Scherzer, O.: Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Probl. 22(3), 801–814 (2006)
Rondi, L.: Uniqueness and stability for the determination of boundary defects by electrostatic measurements. Proc. R. Soc. Edinb. Sect. A 130, 1119–1151 (2000)
Scherzer, O., Engl, H.W., Kunisch, K.: Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J. Numer. Anal. 30(6), 1796–1838 (1993)
Scherzer, O. (ed.): Mathematical Models for Registration and Applications to Medical Imaging. Mathematics in Industry 10. The European Consortium for Mathematics in Industry. Springer, Berlin (2006)
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeiner, M., Lenzen, F.: Variational Methods in Imaging. Springer, New York (2009)
Schleicher, K.-Th., Schulz, S.W., Gmeiner, R., Chun, H.-U.: A computational method for the evaluation of highly resolved DOS functions from APS measurements. J. Electron Spectrosc. Relat. Phenom. 31, 33–56 (1983)
Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10. Walter de Gruyter, Berlin/Boston (2012)
Seidman, T.I., Vogel, C.R.: Well posedness and convergence of some regularization methods for nonlinear ill posed problems. Inverse Probl. 5(2), 227–238 (1989)
Tautenhahn, U.: On a general regularization scheme for nonlinear ill-posed problems. Inverse Probl. 13(5), 1427–1437 (1997)
Tautenhahn, U.: On the method of Lavrentiev regularization for nonlinear ill-posed problems. Inverse Probl. 18(1), 191–207 (2002)
Tautenhahn, U., Jin, Q.: Tikhonov regularization and a posteriori rules for solving nonlinear ill-posed problems. Inverse Probl. 19(1), 1–21 (2003)
Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Dokl. Akad. Nauk SSR 151, 501–504 (1963)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977)
Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht (1995)
Tikhonov, A.N., Leonov, A.S., Yagola, A.G.: Nonlinear Ill-Posed Problems, vols. 1 and 2. Series Applied Mathematics and Mathematical Computation, vol. 14. Chapman & Hall, London (1998)
Vasin, V.V.: Some tendencies in the Tikhonov regularization of ill-posed problems. J. Inverse Ill-Posed Probl. 14(8), 813–840 (2006)
Vogel, C.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)
Werner, F., Hohage, T.: Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data. Inverse Probl. 28(10), 104004(15pp) (2012)
Yamamoto, M.: On ill-posedness and a Tikhonov regularization for a multidimensional inverse hyperbolic problem. J. Math. Kyoto Univ. 36, 825–856 (1996)
Zarzer, C.A.: On Tikhonov regularization with non-convex sparsity constraints. Inverse Probl. 25(2), 025006(13pp) (2009)
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Cheng, J., Hofmann, B. (2014). Regularization Methods for Ill-Posed Problems. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27795-5_3-5
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