Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range and X and Y are Hilbert spaces. The solution x=x † is assumed to be in the range of some selfadjoint strictly positive bounded linear operator G:X →X. Under several assumptions on G, such as or more generally , inequalities of the form , or range inclusions , convergence rates for the regularization error of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The article contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.

[1]  M. Z. Nashed,et al.  A NEW APPROACH TO CLASSIFICATION AND REGULARIZATION OF ILL-POSED OPERATOR EQUATIONS , 1987 .

[2]  Eberhard Schock,et al.  Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence , 1985 .

[3]  J. Weidmann.  Lineare Operatoren in Hilberträumen , 2000 .

[4]  G. Hellwig.  Lineare Operatoren in ℌ , 1964 .

[5]  B. Hofmann.  Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators , 2006 .

[6]  Andreas Neubauer,et al.  When do Sobolev spaces form a Hilbert scale , 1988 .

[7]  R. Bhatia.  Matrix Analysis , 1996 .

[8]  R. Gorenflo,et al.  Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction , 2002 .

[9]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .

[10]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[11]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[12]  Markus Hegland,et al.  Variable hilbert scales and their interpolation inequalities with applications to tikhonov regularization , 1995 .

[13]  Brett Borden,et al.  Ill-Posed Problems , 1999 .

[14]  P. Mathé,et al.  Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 , 2003 .

[15]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[16]  A. Bakushinskii,et al.  Ill-Posed Problems: Theory and Applications , 1994 .

[17]  Andreas Neubauer,et al.  Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales , 1988 .

[18]  Andreas Neubauer,et al.  On Converse and Saturation Results for Tikhonov Regularization of Linear Ill-Posed Problems , 1997 .

[19]  Ulrich Tautenhahn,et al.  Optimality for ill-posed problems under general source conditions , 1998 .

[20]  Ulrich Tautenhahn,et al.  Error Estimates for Regularization Methods in Hilbert Scales , 1996 .

[21]  R. Kress.  Linear Integral Equations , 1989 .

[22]  B. Hofmann,et al.  Some results and a conjecture on the degree of ill-posedness for integration operators with weights , 2005 .

[23]  F. Natterer.  Error bounds for tikhonov regularization in hilbert scales , 1984 .

[24]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[25]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[26]  B. Hofmann.  Regularization for Applied Inverse and III-Posed Problems , 1986 .

[27]  B. Hofmann,et al.  Stability rates for linear ill-posed problems with compact and non-compact operators. , 1999 .

[28]  Peter Mathé,et al.  Regularization of some linear ill-posed problems with discretized random noisy data , 2006, Math. Comput..

[29]  S. Wassermann,et al.  C*‐ALGEBRAS BY EXAMPLE (Fields Institute Monographs 6) , 1998 .

[30]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[31]  Bernard A. Mair,et al.  Tikhonov regularization for finitely and infinitely smoothing operators , 1994 .

[32]  J. Baumeister.  Stable solution of inverse problems , 1987 .

[33]  Sergei V. Pereverzev,et al.  Regularization in Hilbert scales under general smoothing conditions , 2005 .

[34]  Thorsten Hohage,et al.  Regularization of exponentially ill-posed problems , 2000 .

[35]  Otmar Scherzer,et al.  Factors influencing the ill-posedness of nonlinear problems , 1994 .

[36]  K. Davidson.  C*-algebras by example , 1996 .

[37]  Bernd Hofmann,et al.  Convergence rates for Tikhonov regularization based on range inclusions , 2005 .

[38]  P. Mathé,et al.  MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS , 2002 .