## The trade-off between regularity and stability in Tikhonov regularization

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- by M. Thamban Nair, Markus Hegland and Robert S. Anderssen PDF
- Math. Comp.
**66**(1997), 193-206 Request permission

## Abstract:

When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill–posed operator equations, assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it is clear from works of Hegland, Engl and Neubauer and Natterer that, in terms of the rate of convergence, there is a trade–off between stabilization and regularity. It is this matter which is examined in this paper by means of the best–possible worst–error estimates. The results of this paper provide better estimates than those of Engl and Neubauer, and also include and extend the best possible rate derived by Natterer. The paper concludes with an application of these results to first–kind integral equations with smooth kernels.## References

- F. R. de Hoog,
*Review of Fredholm equations of the first kind*, Application and numerical solution of integral equations (Proc. Sem., Australian Nat. Univ., Canberra, 1978) Monographs Textbooks Mech. Solids Fluids: Mech. Anal., vol. 6, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 119–134. MR**582987** - Heinz W. Engl and C. W. Groetsch (eds.),
*Inverse and ill-posed problems*, Notes and Reports in Mathematics in Science and Engineering, vol. 4, Academic Press, Inc., Boston, MA, 1987. Papers from the Alpine-U.S. Seminar held in Sankt Wolfgang, June 1986. MR**1020304** - H.W. Engl and A. Neubauer, Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind, In
*Constructive Methods for the Practical Treatment of Integral Equations*, (G.Hämmerlin and K.-H.Hoffmann, eds.), Birkhäuser, Basel, Boston, Stuttgart, 1985, pp. 120–141. - S. George and M. T. Nair,
*Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates*, J. Optim. Theory Appl.**83**(1994), no. 1, 217–222. MR**1298866**, DOI 10.1007/BF02191771 - S. George and M. T. Nair, On a generalized Arcangeli’s method for Tikhonov regularization with inexact data,
*Research Report, CMA–MR 43–93; SMS–88–93, Australian National University*, 1993. - C. W. Groetsch,
*The theory of Tikhonov regularization for Fredholm equations of the first kind*, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**742928** - Charles W. Groetsch,
*Inverse problems in the mathematical sciences*, Vieweg Mathematics for Scientists and Engineers, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR**1247696**, DOI 10.1007/978-3-322-99202-4 - C. W. Groetsch and J. T. King,
*The saturation phenomena for Tikhonov regularization*, J. Austral. Math. Soc. Ser. A**35**(1983), no. 2, 254–262. MR**704432** - M. Hegland,
*Numerische Lösung von Fredholmschen Integralgleichungen erster Art bei ungenauen Daten*. PhD thesis, ETHZ, 1988. - Markus Hegland,
*An optimal order regularization method which does not use additional smoothness assumptions*, SIAM J. Numer. Anal.**29**(1992), no. 5, 1446–1461. MR**1182739**, DOI 10.1137/0729083 - M. Hegland, Variable Hilbert Scales and their Interpolation Inequalities with Applications to Tikhonov Regularization,
*Applicable Anal.***59**(1995), 207–223. - Tosio Kato,
*Perturbation theory for linear operators*, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR**0407617** - S. G. Kreĭn and Ju. I. Petunin,
*Scales of Banach spaces*, Uspehi Mat. Nauk**21**(1966), no. 2 (128), 89–168 (Russian). MR**0193499** - John Locker and P. M. Prenter,
*Regularization with differential operators. I. General theory*, J. Math. Anal. Appl.**74**(1980), no. 2, 504–529. MR**572669**, DOI 10.1016/0022-247X(80)90145-6 - John Locker and P. M. Prenter,
*Regularization with differential operators. I. General theory*, J. Math. Anal. Appl.**74**(1980), no. 2, 504–529. MR**572669**, DOI 10.1016/0022-247X(80)90145-6 - Jürg T. Marti,
*Numerical solution of Fujita’s equation*, Improperly posed problems and their numerical treatment (Oberwolfach, 1982) Internat. Schriftenreihe Numer. Math., vol. 63, Birkhäuser, Basel, 1983, pp. 179–187. MR**726772** - C. A. Micchelli and T. J. Rivlin,
*A survey of optimal recovery*, Optimal estimation in approximation theory (Proc. Internat. Sympos., Freudenstadt, 1976) Plenum, New York, 1977, pp. 1–54. MR**0617931** - V. A. Morozov,
*Methods for solving incorrectly posed problems*, Springer-Verlag, New York, 1984. Translated from the Russian by A. B. Aries; Translation edited by Z. Nashed. MR**766231**, DOI 10.1007/978-1-4612-5280-1 - M. Thamban Nair,
*A generalization of Arcangeli’s method for ill-posed problems leading to optimal rates*, Integral Equations Operator Theory**15**(1992), no. 6, 1042–1046. MR**1188793**, DOI 10.1007/BF01203127 - Frank Natterer,
*Error bounds for Tikhonov regularization in Hilbert scales*, Applicable Anal.**18**(1984), no. 1-2, 29–37. MR**762862**, DOI 10.1080/00036818408839508 - Andreas Neubauer,
*An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates*, SIAM J. Numer. Anal.**25**(1988), no. 6, 1313–1326. MR**972456**, DOI 10.1137/0725074 - David L. Phillips,
*A technique for the numerical solution of certain integral equations of the first kind*, J. Assoc. Comput. Mach.**9**(1962), 84–97. MR**134481**, DOI 10.1145/321105.321114 - Eberhard Schock,
*Parameter choice by discrepancy principles for the approximate solution of ill-posed problems*, Integral Equations Operator Theory**7**(1984), no. 6, 895–898. MR**774730**, DOI 10.1007/BF01195873 - E. Schock,
*On the asymptotic order of accuracy of Tikhonov regularization*, J. Optim. Theory Appl.**44**(1984), no. 1, 95–104. MR**764866**, DOI 10.1007/BF00934896 - Andrey N. Tikhonov and Vasiliy Y. Arsenin,
*Solutions of ill-posed problems*, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR**0455365** - Manfred R. Trummer,
*A method for solving ill-posed linear operator equations*, SIAM J. Numer. Anal.**21**(1984), no. 4, 729–737. MR**749367**, DOI 10.1137/0721049 - J. M. Varah,
*Pitfalls in the numerical solution of linear ill-posed problems*, SIAM J. Sci. Statist. Comput.**4**(1983), no. 2, 164–176. MR**697171**, DOI 10.1137/0904012

## Additional Information

**M. Thamban Nair**- Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
- Address at time of publication: Department of Mathematics, Indian Institute of Technology, Madras - 600 036, India
- Email: mtnair@acer.iitm.ernet.in
**Markus Hegland**- Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
- Address at time of publication: Computer Sciences Laboratory, RSISE, Australian National University, Canberra ACT 0200, Australia
- Email: Markus.Hegland@anu.edu.au
**Robert S. Anderssen**- Affiliation: Centre for Mathematics and Its Applications, Australian National University, Canberra ACT 0200, Australia
- Address at time of publication: CSIRO Division of Mathematics and Statistics, GPO Box 1965, Canberra ACT 2601, Australia
- Email: boba@cbr.dms.csiro.au
- Received by editor(s): April 13, 1994
- Received by editor(s) in revised form: November 13, 1995
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp.
**66**(1997), 193-206 - MSC (1991): Primary 65R30; Secondary 65J20, 45B05
- DOI: https://doi.org/10.1090/S0025-5718-97-00811-9
- MathSciNet review: 1388891