On the convergence of the hybrid method for an inverse boundary value potential problem in a semi-infinite domain

R.Chapko; N.Vintonyak

We consider a question related to the convergence analysis of the hybrid method used for an inverse potential problem in a semi-infinite region. The local convergence of this method when the data error tends to zero is proved.

1. R. Chapko, R. Kress, A hybrid method for inverse boundary value problems in potential theory, Journal of Ill-Posed and Inverse Problems 13 (2005), 27–-40.

2. R. Chapko, N. Vintonyak, A hybrid method for inverse boundary value problems for an inclusion in semi-infinite two-dimensional domains, Journal of Integral Equations and Applications 19 (2007), 311–-333.

3. R. Chapko, N. Vintonyak, A hybrid method for the solution of inverse potential boundary value problems in semi-infinite domains with crack, Reports of the National Academy of Sciences of Ukraine 7 (2007), 42–-48 (in Ukrainian).

4. R. Chapko, N. Vintonyak, Location search technique and size estimation of the bounded inclusion for some inverse boundary value problems, Visnyk Khmelnytskogo universytetu, 2(93) (2007), №3, 186-191.

5. H.W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Kluwer Academic Publishers, London, 1996.

6. T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems 13 (1997), 1279-1299.

7. V.K. Ivanov, Integral equations of the first kind and an approximate solution for the inverse problem of potential, Soviet Math. Doklady 3 (1962), 210-212.

8. A. Kirsch, R. Kress, An optimization method in inverse acoustic scattering Boundary Element IX, Springer-Verlag, Berlin, 3 (1987), 3-18.

9. R. Kress, Linear Integral Equations, 2nd. ed. Springer-Verlag, Berlin, 1999.

10. R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems 19 (2003), 91-104.

11. R. Kress, N. Vintonyak, Iterative methods for planar crack reconstruction in semi-infinite domains, Journal of Inverse and Ill-Posed Problems 16 (2008), 743-761.

12. R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems 17 (2001), 1419-1434.

13. P. Serranho, A hybrid method for inverse obstacle scattering problems. Dissertation.- Georg-August-Universitaet Goettingen, 2007.

	On the convergence of the hybrid method for an inverse boundary value potential problem in a semi-infinite domain
Author	R.Chapko, N.Vintonyak Ivan Franko National University of Lviv,Faculty of Applied Mathematics and Computer Science
Abstract	We consider a question related to the convergence analysis of the hybrid method used for an inverse potential problem in a semi-infinite region. The local convergence of this method when the data error tends to zero is proved.
Keywords	hybrid method, inverse boundary value potential problem, semi-inifnite domain
DOI	doi:10.30970/ms.32.1.45-52
Reference	1. R. Chapko, R. Kress, A hybrid method for inverse boundary value problems in potential theory, Journal of Ill-Posed and Inverse Problems 13 (2005), 27–-40. 2. R. Chapko, N. Vintonyak, A hybrid method for inverse boundary value problems for an inclusion in semi-infinite two-dimensional domains, Journal of Integral Equations and Applications 19 (2007), 311–-333. 3. R. Chapko, N. Vintonyak, A hybrid method for the solution of inverse potential boundary value problems in semi-infinite domains with crack, Reports of the National Academy of Sciences of Ukraine 7 (2007), 42–-48 (in Ukrainian). 4. R. Chapko, N. Vintonyak, Location search technique and size estimation of the bounded inclusion for some inverse boundary value problems, Visnyk Khmelnytskogo universytetu, 2(93) (2007), №3, 186-191. 5. H.W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Kluwer Academic Publishers, London, 1996. 6. T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems 13 (1997), 1279-1299. 7. V.K. Ivanov, Integral equations of the first kind and an approximate solution for the inverse problem of potential, Soviet Math. Doklady 3 (1962), 210-212. 8. A. Kirsch, R. Kress, An optimization method in inverse acoustic scattering Boundary Element IX, Springer-Verlag, Berlin, 3 (1987), 3-18. 9. R. Kress, Linear Integral Equations, 2nd. ed. Springer-Verlag, Berlin, 1999. 10. R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems 19 (2003), 91-104. 11. R. Kress, N. Vintonyak, Iterative methods for planar crack reconstruction in semi-infinite domains, Journal of Inverse and Ill-Posed Problems 16 (2008), 743-761. 12. R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems 17 (2001), 1419-1434. 13. P. Serranho, A hybrid method for inverse obstacle scattering problems. Dissertation.- Georg-August-Universitaet Goettingen, 2007.
Pages	45-52
Volume	32
Issue	1
Year	2009
Journal	Matematychni Studii
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