Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

References

[1] A. B. Bakushinskiĭ, On a convergence problem of the iterative-regularized Gauss–Newton method, Comput. Math. Math. Phys. 32 (1992), no. 9, 1503–1509. Search in Google Scholar

[2] A. B. Bakushinskiĭ, Iterative methods without saturation for solving degenerate nonlinear operator equations, Dokl. Akad. Nauk 344 (1995), no. 1, 7–8. Search in Google Scholar

[3] H. Egger, Preconditioning Iterative Regularization Methods in Hilbert scales, Dissertation, Johannes Kepler Universität, Linz, 2005. Search in Google Scholar

[4] H. Egger, Semi-iterative regularization in Hilbert scales, SIAM J. Numer. Anal. 44 (2006), no. 1, 66–81. 10.1137/040617285Search in Google Scholar

[5] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic Publishers, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar

[6] S. George and M. T. Nair, Error bounds and parameter choice strategies for simplified regularization in Hilbert scales, Integral Equations Operator Theory 29 (1997), no. 2, 231–242. 10.1007/BF01191432Search in Google Scholar

[7] 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), no. 5, 1279–1299. 10.1088/0266-5611/13/5/012Search in Google Scholar

[8] Q. Jin, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse Problem 16 (1999), 187–197. 10.1088/0266-5611/16/1/315Search in Google Scholar

[9] Q. Jin, Further convergence results on the general iteratively regularized Gauss–Newton methods under the discrepancy principle, Math. Comp. 82 (2013), no. 283, 1647–1665. 10.1090/S0025-5718-2012-02665-2Search in Google Scholar

[10] Q. Jin and U. Tautenhahn, On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems, Numer. Math. 111 (2009), no. 4, 509–558. 10.1007/s00211-008-0198-ySearch in Google Scholar

[11] B. Kaltenbacher, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math. 79 (1998), no. 4, 501–528. 10.1007/s002110050349Search in Google Scholar

[12] C. Kravaris and J. H. Seinfeld, Identification of parameters in distributed parameter systems by regularization, SIAM J. Control Optim. 23 (1985), no. 2, 217–241. 10.1109/CDC.1983.269793Search in Google Scholar

[13] S. Lu, S. V. Pereverzev, Y. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, J. Integral Equations Appl. 22 (2010), no. 3, 483–517. 10.1216/JIE-2010-22-3-483Search in Google Scholar

[14] P. Mahale and M. T. Nair, A simplified generalized Gauss-Newton method for nonlinear ill-posed problems, Math. Comp. 78 (2009), no. 265, 171–184. 10.1090/S0025-5718-08-02149-2Search in Google Scholar

[15] M. T. Nair, Role of Hilbert scales in regularization theory, Semigroups, Algebras and Operator Theory, Springer Proc. Math. Stat. 142, Sprinhger, Cham (2015), 159–176. 10.1007/978-81-322-2488-4_13Search in Google Scholar

[16] M. T. Nair, Compact operators and Hilbert scales in ill-posed problems, Math. Student 85 (2016), 45–61. Search in Google Scholar

[17] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal. 18 (1984), no. 1–2, 29–37. 10.1080/00036818408839508Search in Google Scholar

[18] A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales, Appl. Anal. 46 (1992), no. 1–2, 59–72. 10.1080/00036819208840111Search in Google Scholar

[19] U. Tautenhahn, Error estimates for regularization methods in Hilbert scales, SIAM J. Numer. Anal. 33 (1996), no. 6, 2120–2130. 10.1137/S0036142994269411Search in Google Scholar

[20] U. Tautenhahn, On a general regularization scheme for nonlinear ill-posed problems. II. Regularization in Hilbert scales, Inverse Problems 14 (1998), no. 6, 1607–1616. 10.1088/0266-5611/14/6/016Search in Google Scholar

[21] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, V. H. Winston & Sons, Washington, 1977. Search in Google Scholar