Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

References

• Ang , DD , Gorenflo , R , Le , VK and Trong , DD . 2002 . Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction , Berlin : Springer . Lect. Notes Math. 1792
   Google Scholar
• Bakushinsky , A and Goncharsky , A . 1994 . Ill-Posed Problems: Theory and Applications , Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Baumeister , J . 1987 . Stable Solution of Inverse Problems , Braunschweig : Vieweg .
   Google Scholar
• Bhatia , R . 1997 . Matrix Analysis , New York : Springer .
   Google Scholar
• Davidson , KR . 1996 . C*-Algebras by Example , Fields Institute Monographs, Providence : American Mathematical Society .
   Google Scholar
• Engl , HW , Hanke , M and Neubauer , A . 1996 . Regularization of Inverse Problems , Dordrecht : Kluwer .
   Google Scholar
• Groetsch , CW . 1984 . The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind , Boston : Pitman .
   Google Scholar
• Hegland , M . 1995 . Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization . Applicable Analysis , 59 : 207 – 223 .
   Google Scholar
• Hofmann , B . 1986 . Regularization of Applied Inverse and Ill-Posed Problems , Leipzig : Teubner .
   Google Scholar
• Hofmann , B . 2006 . Approximate source conditions in Tikhonov-Phillips regularization and consequences for inverse problems with multiplication operators . Mathematical Methods in the Applied Sciences , 29 (in press – published online 2005 as DOI: 10.1002/mma686)
   Google Scholar
• Hofmann , B and Fleischer , G . 1999 . Stability rates for linear ill-posed problems with compact and non-compact operators . Zeitschrift fur Analysis und ihre Anwendungen , 18 : 267 – 286 .
   Web of Science ®Google Scholar
• Hofmann , B and Scherzer , O . 1994 . Factors influencing the ill-posedness of nonlinear problems . Inverse Problems , 10 : 1277 – 1297 .
   Web of Science ®Google Scholar
• Hofmann , B and von Wolfersdorf , L . 2005 . Some results and a conjecture on the degree of ill-posedness for integration operators with weights . Inverse Problems , 21 : 427 – 433 .
   Web of Science ®Google Scholar
• Hofmann , B and Yamamoto , M . 2005 . Convergence rates for Tikhonov regularization based on range inclusions . Inverse Problems , 21 : 805 – 820 .
   Web of Science ®Google Scholar
• Hohage , T . 2000 . Regularization of exponentially ill-posed problems . Numerical Functional Analysis and Optimization , 21 : 439 – 464 .
   Web of Science ®Google Scholar
• Kirsch , A . 1996 . An Introduction to the Mathematical Theory of Inverse Problems , New York : Springer .
   Google Scholar
• Kress , R . 1989 . Linear Integral Equations , Berlin : Springer .
   Google Scholar
• Louis , AK . 1989 . Inverse und schlecht gestellte Probleme , Stuttgart : Teubner .
   Google Scholar
• Mair , BA . 1994 . Tikhonov regularization for finitely and infinitely smoothing operators . SIAM Journal on Mathematical Analysis , 25 : 135 – 147 .
   Web of Science ®Google Scholar
• Mathé , P and Pereverzev , SV . 2002 . Moduli of continuity for operator valued functions . Numerical Functional Analysis and Optimization , 23 : 623 – 631 .
   Web of Science ®Google Scholar
• Mathé , P and Pereverzev , SV . 2003 . Geometry of linear ill-posed problems in variable Hilbert scales . Inverse Problems , 19 : 789 – 803 .
   Web of Science ®Google Scholar
• Mathé , P and Pereverzev , SV . 2006 . Regularization of some linear ill-posed problems with discretized random noisy data . Mathematics of Computation , 75 (in press)
   Google Scholar
• Morozov , VÅ . 1966 . On the solution of functional equations by the method of regularization . Soviet Mathematics Doklady , 7 : 414 – 417 .
   Google Scholar
• Nair , MT , Pereverzev , SV and Tautenhahn , U . 2005 . Regularization in Hilbert scales under general smoothing conditions . Inverse Problems , 21 : 1851 – 1869 .
   Web of Science ®Google Scholar
• Natterer , F . 1984 . Error bounds for Tikhonov regularization in Hilbert scales . Applicable Analysis , 18 : 29 – 37 .
   Web of Science ®Google Scholar
• Nashed , MZ . 1987 . “ A new approach to classification and regularization of ill-posed operator equations ” . In Inverse and Ill-posed Problems , Edited by: Engl , HW and Groetsch , CW . 53 – 75 . Orlando : Academic Press .
   Google Scholar
• Neubauer , A . 1988 . When do Sobolev spaces form a Hilbert scale? . Proceeding of the American Mathematical Society , 102 : 587 – 592 .
   Google Scholar
• Neubauer , A . 1997 . On converse and saturation results for Tikhonov regularization of linear ill-posed problems . SIAM Journal on Numerical Analysis , 34 : 517 – 527 .
   Web of Science ®Google Scholar
• Schock , E . 1985 . “ Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence ” . In Constructive Methods for the Practical Treatment of Integral Equations , Edited by: Hämmerlin , G and Hoffmann , KH . 234 – 243 . Basel : Birkhäuser .
   Google Scholar
• Tanabe , H . 1979 . Equations of Evolution , London : Pitman .
   Google Scholar
• Tautenhahn , U . 1996 . Error estimates for regularization methods in Hilbert scales . SIAM Journal on Numerical Analysis , 33 : 2120 – 2130 .
   Web of Science ®Google Scholar
• Tautenhahn , U . 1998 . Optimality for ill-posed problems under general source conditions . Numerical Functional Analysis and Optimization , 19 : 377 – 398 .
   Web of Science ®Google Scholar
• Tikhonov , AN . 1963 . Solution of incorrectly formulated problems and the regularization method . Soviet Mathematics Doklady , 4 : 1035 – 1038 .
   Google Scholar
• Tikhonov , AN and Arsenin , VY . 1977 . Solution of Ill-Posed Problems , New York : Wiley .
   Google Scholar
• Weidmann , J . 2000 . Lineare Operatoren in Hilberträumen, Vol. I, Grundlagen , Stuttgart : Teubner .
   Google Scholar
• Vainikko , GM and Veretennikov , AY . 1986 . Iteration Procedures in Ill-posed Problems , Moscow : Nauka . (In Russian)
   Google Scholar