Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales

Authors: Heinz W. Engl and Andreas Neubauer
Journal: Proc. Amer. Math. Soc. 102 (1988), 587-592
MSC: Primary 65J10; Secondary 47A50
MathSciNet review: 928985
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main result of this paper states how the discretization parameter and regularization parameter should be chosen in relation to the noise level in order to yield the optimal convergence rate for the Tikhonov-regularized solution of an ill-posed linear operator equation in a finite-dimensional subspace in the framework of Hilbert scales. The results apply to a wide class of spline and finite-element subspaces of Sobolev scales.

References [Enhancements On Off] (What's this?)

  • [1] I. Babuška and A. Aziz, Survey lectures on the mathematical foundation of the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. Aziz., ed.), Academic Press, New York, 1972, pp. 3-359.
  • [2] J. H. Bramble and Ridgway Scott, Simultaneous approximation in scales of Banach spaces, Math. Comp. 32 (1978), 947-954. MR 501990 (80a:65222)
  • [3] H. W. Engl and A. Neubauer, Optimal discrepancy principles for the Tikhonov-regularization of integral equations of the first kind, Constructive Methods for the Practical Treatment of Integral Equations (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1985, pp. 120-141. MR 882562
  • [4] C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Pitman, Boston, 1984. MR 742928 (85k:45020)
  • [5] S. G. Krein and J. I. Petunin, Scales of Banach spaces, Russian Math. Surveys 21 (1966), 85-160. MR 0193499 (33:1719)
  • [6] J. L. Lions and E. Magenes, Nonhomogenous boundary value problems and applications, vol. I, Springer-Verlag, Berlin and New York, 1972.
  • [7] F. Natterer, Regularisierung schlecht gestellter Probleme durch Projektionsverfahren, Numer. Math. 28 (1977), 329-341. MR 0488721 (58:8238)
  • [8] -, The finite element method for ill-posed problems, RAIRO Anal. Numér 11 (1977), 271-278. MR 0519587 (58:24920)
  • [9] -, On the order of regularization methods, Improperly Posed Problems and Their Numerical Treatment (G. Hämmerlin and K. H. Hoffmann, eds.), Birkhäuser, Basel, 1983, pp. 189-203. MR 726773 (85h:65127)
  • [10] -, Error bounds for Tikhonov regularization in Hilbert scales, Applicable Anal. 18 (1984), 29-37. MR 762862 (86e:65081)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 65J10, 47A50

Retrieve articles in all journals with MSC: 65J10, 47A50

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society