Abstract
This chapter will continue the study of Tikhonov regularization and will be based on its classical interpretation as a penalized residual minimization. For this we will consider the more general case of merely bounded linear operators. In particular, we shall explain the concepts of quasi-solutions and minimum norm solutions as strategies for the selection of the regularization parameter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Christiansen, S.: Condition number of matrices derived from two classes of integral equations. Math. Meth. in the Appl. Sci. 3, 364–392 (1981).
Colton, D., and Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. Springer, Berlin 2013.
Fairweather, G., and Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998).
Golberg, M.A., and Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Boundary Integral Methods – Numerical and Mathematical Aspects (Golberg, ed.). Computational Mechanics Publications, Southampton, 103–176 (1998).
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston 1984.
Ivanov, V.K.: Integral equations of the first kind and an approximate solution for the inverse problem of potential. Soviet Math. Doklady 3, 210–212 (1962) (English translation).
Kress, R., and Mohsen, A.: On the simulation source technique for exterior problems in acoustics. Math. Meth. in the Appl. Sci. 8, 585–597 (1986).
Kupradze, V.D.: Dynamical problems in elasticity. In: Progress in Solid Mechanics, Vol. III (Sneddon and Hill, eds.). North Holland, Amsterdam, 1–259 (1963).
Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Doklady 7, 414–417 (1966) (English translation).
Morozov, V.A.: Choice of parameter for the solution of functional equations by the regularization method. Soviet Math. Doklady 8, 1000–1003 (1967) (English translation).
Phillips, D.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Ass. Comp. Math. 9, 84–97 (1962).
Tikhonov, A.N.: On the solution of incorrectly formulated problems and the regularization method. Soviet Math. Doklady 4, 1035–1038 (1963) (English translation).
Tikhonov, A.N.: Regularization of incorrectly posed problems. Soviet Math. Doklady 4, 1624–1627 (1963) (English translation).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kress, R. (2014). Tikhonov Regularization. In: Linear Integral Equations. Applied Mathematical Sciences, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9593-2_16
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9593-2_16
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9592-5
Online ISBN: 978-1-4614-9593-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)