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August  2011, 5(3): 619-643. doi: 10.3934/ipi.2011.5.619

## Errors of regularisation under range inclusions using variable Hilbert scales

 1 Centre for Mathematics and its Applications, The Australian National University, Canberra ACT, 0200, Australia 2 Department of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany

Received  May 2010 Revised  June 2011 Published  August 2011

Based on the variable Hilbert scale interpolation inequality, bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators with non-closed range are given. Even if one can show the equivalence of this approach to the version used previously in the literature, the new formulae and corresponding conditions are simpler than the former ones. Several examples from image processing and spectral enhancement illustrate how the new error bounds can be applied.
Citation: Markus Hegland, Bernd Hofmann. Errors of regularisation under range inclusions using variable Hilbert scales. Inverse Problems and Imaging, 2011, 5 (3) : 619-643. doi: 10.3934/ipi.2011.5.619
##### References:
 [1] K. Atkinson and W. Han, "Theoretical Numerical Analysis. A Functional Analysis Framework," 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. [2] A. B. Bakushinsky and M. Yu. Kokurin, "Iterative Methods for Approximate Solution of Inverse Problems," Mathematics and its Applications (New York), 577, Springer, Dordrecht, 2004. [3] J. Baumeister, "Stable Solution of Inverse Problems," Friedr. Vieweg & Sohn, Braunschweig, 1987. [4] M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging," Institute of Physics Publishing, Bristol, 1998. [5] R. Bhatia, "Matrix Analysis," Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. [6] A. Böttcher, B. Hofmann, U. Tautenhahn and M. Yamamoto, Convergence rates for Tikhonov regularization from different kinds of smoothness conditions, Appl. Anal., 85 (2006), 555-578. [7] J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization, Inverse Problems, 16 (2000), L31-L38. [8] D. Düvelmeyer, B. Hofmann and M. Yamamoto, Range inclusions and approximate source conditions with general benchmark functions, Numer. Funct. Anal. Optim., 28 (2007), 1245-1261. [9] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. [10] J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems, 27 (2011), 025006 (18 pp). [11] C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind," Pitman, Boston, MA, 1984. [12] M. Hanke, Regularization with differential operators: An iterative approach, Numer. Funct. Anal. & Optimiz., 13 (1992), 523-540. doi: 10.1080/01630569208816497. [13] M. Hegland, An optimal order regularization method which does not use additional smoothness assumptions, SIAM J. Numer. Anal., 29 (1992), 1446-1461. [14] _____, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal., 59 (1995), 207-223. [15] _____, Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities, J. Integral Equations Appl., 22 (2010), 285-312. [16] M. Hegland and B. Hofmann, Errors of regularisation under range inclusions using variable Hilbert scales,, 2010. Available from: \url{http://arxiv.org/abs/1005.3883}., (). [17] B. Hofmann, Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators, Mathematical Methods in the Applied Sciences, 29 (2006), 351-371. doi: 10.1002/mma.686. [18] B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods, SIAM J. Numer. Anal., 45 (2007), 1122-1141. [19] B. Hofmann, P. Mathé and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, J. Inverse Ill-Posed Probl., 16 (2008), 567-585. [20] B. Hofmann, P. Mathé and H. von Weizsäcker, Regularization in Hilbert space under unbounded operators and general source conditions, Inverse Problems, 25 (2009), 115013 (15 pp). [21] B. Hofmann and M. Yamamoto, Convergence rates for Tikhonov regularization based on range inclusions, Inverse Problems, 21 (2005), 805-820. [22] T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299. [23] _____, Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim., 21 (2000), 439-464. [24] V. K. Ivanov and T. I. Koroljuk, The estimation of errors in the solution of linear ill-posed problems, Ž. Vyčisl. Mat. i Mat. Fiz., 9 (1969), 30-41. [25] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. [26] A. K. Louis, "Inverse und Schlecht Gestellte Probleme," (German) [Inverse and ill-posed problems], Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, 1989. [27] B. Mair, Tikhonov regularization for finitely and infinitely smoothing operators, SIAM J. Math. Anal., 25 (1994), 135-147. [28] P. Mathé and S. V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 1263-1277. [29] _____, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803. [30] P. Mathé and U. Tautenhahn, Interpolation in variable Hilbert scales with application to inverse problems, Inverse Problems, 22 (2006), 2271-2297. [31] M. T. Nair, E. Schock and U. Tautenhahn, Morozov's discrepancy principle under general source conditions, Z. Anal. Anwendungen, 22 (2003), 199-214. [32] M. T. Nair, S. Pereverzev and U. Tautenhahn, Regularization in Hilbert scales under general smoothing conditions, Inverse Problems, 21 (2005), 1851-1869. [33] M. T. Nair, "Linear Operator Equations: Approximation and Regularization," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. doi: 10.1142/9789812835659. [34] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18 (1984), 29-37. [35] M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis," Second edition, Academic Press Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [36] A. Rieder, "Keine Probleme mit inversen Problemen," (German) [No Problems with Inverse Problems], Eine Einführung in ihre stabile Lösung [An Introduction to their Stable Solution], Friedr. Vieweg & Sohn, Braunschweig, 2003. [37] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Applied Mathematical Sciences, 167, Springer, New York, 2009. [38] E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence, in "Constructive Methods for the Practical Treatment of Integral Equations" (eds. G. Hämmerlin and K. H. Hoffmann), Internat. Schriftenreihe Numer.Math., 73, Birkhäuser, Basel, (1985), 234-243. [39] F. R. Stauffer and H. Sakai, Derivative spectroscopy, Applied Optics, 7 (1968), 61-65. [40] U. Tautenhahn, Error estimates for regularization methods in Hilbert scales, SIAM J. Numer. Anal., 33 (1996), 2120-2130. [41] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. & Optimiz., 19 (1998), 377-398. doi: 10.1080/01630569808816834. [42] D. Werner, "Funktionalanalysis," Third, revised and extended ed., Springer-Verlag, Berlin, 2000.

show all references

##### References:
 [1] K. Atkinson and W. Han, "Theoretical Numerical Analysis. A Functional Analysis Framework," 3rd edition, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. [2] A. B. Bakushinsky and M. Yu. Kokurin, "Iterative Methods for Approximate Solution of Inverse Problems," Mathematics and its Applications (New York), 577, Springer, Dordrecht, 2004. [3] J. Baumeister, "Stable Solution of Inverse Problems," Friedr. Vieweg & Sohn, Braunschweig, 1987. [4] M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging," Institute of Physics Publishing, Bristol, 1998. [5] R. Bhatia, "Matrix Analysis," Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. [6] A. Böttcher, B. Hofmann, U. Tautenhahn and M. Yamamoto, Convergence rates for Tikhonov regularization from different kinds of smoothness conditions, Appl. Anal., 85 (2006), 555-578. [7] J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization, Inverse Problems, 16 (2000), L31-L38. [8] D. Düvelmeyer, B. Hofmann and M. Yamamoto, Range inclusions and approximate source conditions with general benchmark functions, Numer. Funct. Anal. Optim., 28 (2007), 1245-1261. [9] H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. [10] J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions, Inverse Problems, 27 (2011), 025006 (18 pp). [11] C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind," Pitman, Boston, MA, 1984. [12] M. Hanke, Regularization with differential operators: An iterative approach, Numer. Funct. Anal. & Optimiz., 13 (1992), 523-540. doi: 10.1080/01630569208816497. [13] M. Hegland, An optimal order regularization method which does not use additional smoothness assumptions, SIAM J. Numer. Anal., 29 (1992), 1446-1461. [14] _____, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal., 59 (1995), 207-223. [15] _____, Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities, J. Integral Equations Appl., 22 (2010), 285-312. [16] M. Hegland and B. Hofmann, Errors of regularisation under range inclusions using variable Hilbert scales,, 2010. Available from: \url{http://arxiv.org/abs/1005.3883}., (). [17] B. Hofmann, Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators, Mathematical Methods in the Applied Sciences, 29 (2006), 351-371. doi: 10.1002/mma.686. [18] B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods, SIAM J. Numer. Anal., 45 (2007), 1122-1141. [19] B. Hofmann, P. Mathé and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, J. Inverse Ill-Posed Probl., 16 (2008), 567-585. [20] B. Hofmann, P. Mathé and H. von Weizsäcker, Regularization in Hilbert space under unbounded operators and general source conditions, Inverse Problems, 25 (2009), 115013 (15 pp). [21] B. Hofmann and M. Yamamoto, Convergence rates for Tikhonov regularization based on range inclusions, Inverse Problems, 21 (2005), 805-820. [22] T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299. [23] _____, Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim., 21 (2000), 439-464. [24] V. K. Ivanov and T. I. Koroljuk, The estimation of errors in the solution of linear ill-posed problems, Ž. Vyčisl. Mat. i Mat. Fiz., 9 (1969), 30-41. [25] A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996. [26] A. K. Louis, "Inverse und Schlecht Gestellte Probleme," (German) [Inverse and ill-posed problems], Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, 1989. [27] B. Mair, Tikhonov regularization for finitely and infinitely smoothing operators, SIAM J. Math. Anal., 25 (1994), 135-147. [28] P. Mathé and S. V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 1263-1277. [29] _____, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 789-803. [30] P. Mathé and U. Tautenhahn, Interpolation in variable Hilbert scales with application to inverse problems, Inverse Problems, 22 (2006), 2271-2297. [31] M. T. Nair, E. Schock and U. Tautenhahn, Morozov's discrepancy principle under general source conditions, Z. Anal. Anwendungen, 22 (2003), 199-214. [32] M. T. Nair, S. Pereverzev and U. Tautenhahn, Regularization in Hilbert scales under general smoothing conditions, Inverse Problems, 21 (2005), 1851-1869. [33] M. T. Nair, "Linear Operator Equations: Approximation and Regularization," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. doi: 10.1142/9789812835659. [34] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18 (1984), 29-37. [35] M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis," Second edition, Academic Press Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [36] A. Rieder, "Keine Probleme mit inversen Problemen," (German) [No Problems with Inverse Problems], Eine Einführung in ihre stabile Lösung [An Introduction to their Stable Solution], Friedr. Vieweg & Sohn, Braunschweig, 2003. [37] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging," Applied Mathematical Sciences, 167, Springer, New York, 2009. [38] E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence, in "Constructive Methods for the Practical Treatment of Integral Equations" (eds. G. Hämmerlin and K. H. Hoffmann), Internat. Schriftenreihe Numer.Math., 73, Birkhäuser, Basel, (1985), 234-243. [39] F. R. Stauffer and H. Sakai, Derivative spectroscopy, Applied Optics, 7 (1968), 61-65. [40] U. Tautenhahn, Error estimates for regularization methods in Hilbert scales, SIAM J. Numer. Anal., 33 (1996), 2120-2130. [41] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. & Optimiz., 19 (1998), 377-398. doi: 10.1080/01630569808816834. [42] D. Werner, "Funktionalanalysis," Third, revised and extended ed., Springer-Verlag, Berlin, 2000.
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