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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 & 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, 39 (2009).   Google Scholar

[2]

A. B. Bakushinsky and M. Yu. Kokurin, "Iterative Methods for Approximate Solution of Inverse Problems,", Mathematics and its Applications (New York), 577 (2004).   Google Scholar

[3]

J. Baumeister, "Stable Solution of Inverse Problems,", Friedr. Vieweg & Sohn, (1987).   Google Scholar

[4]

M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging,", Institute of Physics Publishing, (1998).   Google Scholar

[5]

R. Bhatia, "Matrix Analysis,", Graduate Texts in Mathematics, 169 (1997).   Google Scholar

[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.   Google Scholar

[7]

J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization,, Inverse Problems, 16 (2000).   Google Scholar

[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.   Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Mathematics and its Applications, 375 (1996).   Google Scholar

[10]

J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions,, Inverse Problems, 27 (2011).   Google Scholar

[11]

C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind,", Pitman, (1984).   Google Scholar

[12]

M. Hanke, Regularization with differential operators: An iterative approach,, Numer. Funct. Anal. & Optimiz., 13 (1992), 523.  doi: 10.1080/01630569208816497.  Google Scholar

[13]

M. Hegland, An optimal order regularization method which does not use additional smoothness assumptions,, SIAM J. Numer. Anal., 29 (1992), 1446.   Google Scholar

[14]

_____, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization,, Appl. Anal., 59 (1995), 207.   Google Scholar

[15]

_____, Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities,, J. Integral Equations Appl., 22 (2010), 285.   Google Scholar

[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}., ().   Google Scholar

[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.  doi: 10.1002/mma.686.  Google Scholar

[18]

B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods,, SIAM J. Numer. Anal., 45 (2007), 1122.   Google Scholar

[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.   Google Scholar

[20]

B. Hofmann, P. Mathé and H. von Weizsäcker, Regularization in Hilbert space under unbounded operators and general source conditions,, Inverse Problems, 25 (2009).   Google Scholar

[21]

B. Hofmann and M. Yamamoto, Convergence rates for Tikhonov regularization based on range inclusions,, Inverse Problems, 21 (2005), 805.   Google Scholar

[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.   Google Scholar

[23]

_____, Regularization of exponentially ill-posed problems,, Numer. Funct. Anal. Optim., 21 (2000), 439.   Google Scholar

[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.   Google Scholar

[25]

A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems,", Applied Mathematical Sciences, 120 (1996).   Google Scholar

[26]

A. K. Louis, "Inverse und Schlecht Gestellte Probleme,", (German) [Inverse and ill-posed problems], (1989).   Google Scholar

[27]

B. Mair, Tikhonov regularization for finitely and infinitely smoothing operators,, SIAM J. Math. Anal., 25 (1994), 135.   Google Scholar

[28]

P. Mathé and S. V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 1263.   Google Scholar

[29]

_____, Geometry of linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 789.   Google Scholar

[30]

P. Mathé and U. Tautenhahn, Interpolation in variable Hilbert scales with application to inverse problems,, Inverse Problems, 22 (2006), 2271.   Google Scholar

[31]

M. T. Nair, E. Schock and U. Tautenhahn, Morozov's discrepancy principle under general source conditions,, Z. Anal. Anwendungen, 22 (2003), 199.   Google Scholar

[32]

M. T. Nair, S. Pereverzev and U. Tautenhahn, Regularization in Hilbert scales under general smoothing conditions,, Inverse Problems, 21 (2005), 1851.   Google Scholar

[33]

M. T. Nair, "Linear Operator Equations: Approximation and Regularization,", World Scientific Publishing Co. Pte. Ltd., (2009).  doi: 10.1142/9789812835659.  Google Scholar

[34]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales,, Appl. Anal., 18 (1984), 29.   Google Scholar

[35]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis,", Second edition, (1980).   Google Scholar

[36]

A. Rieder, "Keine Probleme mit inversen Problemen,", (German) [No Problems with Inverse Problems], (2003).   Google Scholar

[37]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging,", Applied Mathematical Sciences, 167 (2009).   Google Scholar

[38]

E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence,, in, 73 (1985), 234.   Google Scholar

[39]

F. R. Stauffer and H. Sakai, Derivative spectroscopy,, Applied Optics, 7 (1968), 61.   Google Scholar

[40]

U. Tautenhahn, Error estimates for regularization methods in Hilbert scales,, SIAM J. Numer. Anal., 33 (1996), 2120.   Google Scholar

[41]

U. Tautenhahn, Optimality for ill-posed problems under general source conditions,, Numer. Funct. Anal. & Optimiz., 19 (1998), 377.  doi: 10.1080/01630569808816834.  Google Scholar

[42]

D. Werner, "Funktionalanalysis,", Third, (2000).   Google Scholar

show all references

References:
[1]

K. Atkinson and W. Han, "Theoretical Numerical Analysis. A Functional Analysis Framework,", 3rd edition, 39 (2009).   Google Scholar

[2]

A. B. Bakushinsky and M. Yu. Kokurin, "Iterative Methods for Approximate Solution of Inverse Problems,", Mathematics and its Applications (New York), 577 (2004).   Google Scholar

[3]

J. Baumeister, "Stable Solution of Inverse Problems,", Friedr. Vieweg & Sohn, (1987).   Google Scholar

[4]

M. Bertero and P. Boccacci, "Introduction to Inverse Problems in Imaging,", Institute of Physics Publishing, (1998).   Google Scholar

[5]

R. Bhatia, "Matrix Analysis,", Graduate Texts in Mathematics, 169 (1997).   Google Scholar

[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.   Google Scholar

[7]

J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization,, Inverse Problems, 16 (2000).   Google Scholar

[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.   Google Scholar

[9]

H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Mathematics and its Applications, 375 (1996).   Google Scholar

[10]

J. Flemming, B. Hofmann and P. Mathé, Sharp converse results for the regularization error using distance functions,, Inverse Problems, 27 (2011).   Google Scholar

[11]

C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind,", Pitman, (1984).   Google Scholar

[12]

M. Hanke, Regularization with differential operators: An iterative approach,, Numer. Funct. Anal. & Optimiz., 13 (1992), 523.  doi: 10.1080/01630569208816497.  Google Scholar

[13]

M. Hegland, An optimal order regularization method which does not use additional smoothness assumptions,, SIAM J. Numer. Anal., 29 (1992), 1446.   Google Scholar

[14]

_____, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization,, Appl. Anal., 59 (1995), 207.   Google Scholar

[15]

_____, Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities,, J. Integral Equations Appl., 22 (2010), 285.   Google Scholar

[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}., ().   Google Scholar

[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.  doi: 10.1002/mma.686.  Google Scholar

[18]

B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods,, SIAM J. Numer. Anal., 45 (2007), 1122.   Google Scholar

[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.   Google Scholar

[20]

B. Hofmann, P. Mathé and H. von Weizsäcker, Regularization in Hilbert space under unbounded operators and general source conditions,, Inverse Problems, 25 (2009).   Google Scholar

[21]

B. Hofmann and M. Yamamoto, Convergence rates for Tikhonov regularization based on range inclusions,, Inverse Problems, 21 (2005), 805.   Google Scholar

[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.   Google Scholar

[23]

_____, Regularization of exponentially ill-posed problems,, Numer. Funct. Anal. Optim., 21 (2000), 439.   Google Scholar

[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.   Google Scholar

[25]

A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems,", Applied Mathematical Sciences, 120 (1996).   Google Scholar

[26]

A. K. Louis, "Inverse und Schlecht Gestellte Probleme,", (German) [Inverse and ill-posed problems], (1989).   Google Scholar

[27]

B. Mair, Tikhonov regularization for finitely and infinitely smoothing operators,, SIAM J. Math. Anal., 25 (1994), 135.   Google Scholar

[28]

P. Mathé and S. V. Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 1263.   Google Scholar

[29]

_____, Geometry of linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 789.   Google Scholar

[30]

P. Mathé and U. Tautenhahn, Interpolation in variable Hilbert scales with application to inverse problems,, Inverse Problems, 22 (2006), 2271.   Google Scholar

[31]

M. T. Nair, E. Schock and U. Tautenhahn, Morozov's discrepancy principle under general source conditions,, Z. Anal. Anwendungen, 22 (2003), 199.   Google Scholar

[32]

M. T. Nair, S. Pereverzev and U. Tautenhahn, Regularization in Hilbert scales under general smoothing conditions,, Inverse Problems, 21 (2005), 1851.   Google Scholar

[33]

M. T. Nair, "Linear Operator Equations: Approximation and Regularization,", World Scientific Publishing Co. Pte. Ltd., (2009).  doi: 10.1142/9789812835659.  Google Scholar

[34]

F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales,, Appl. Anal., 18 (1984), 29.   Google Scholar

[35]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics I: Functional Analysis,", Second edition, (1980).   Google Scholar

[36]

A. Rieder, "Keine Probleme mit inversen Problemen,", (German) [No Problems with Inverse Problems], (2003).   Google Scholar

[37]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, "Variational Methods in Imaging,", Applied Mathematical Sciences, 167 (2009).   Google Scholar

[38]

E. Schock, Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence,, in, 73 (1985), 234.   Google Scholar

[39]

F. R. Stauffer and H. Sakai, Derivative spectroscopy,, Applied Optics, 7 (1968), 61.   Google Scholar

[40]

U. Tautenhahn, Error estimates for regularization methods in Hilbert scales,, SIAM J. Numer. Anal., 33 (1996), 2120.   Google Scholar

[41]

U. Tautenhahn, Optimality for ill-posed problems under general source conditions,, Numer. Funct. Anal. & Optimiz., 19 (1998), 377.  doi: 10.1080/01630569808816834.  Google Scholar

[42]

D. Werner, "Funktionalanalysis,", Third, (2000).   Google Scholar

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