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Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

Santhosh George ORCID logo and K. Kanagaraj

Abstract

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

Funding statement: The work of Santhosh George is supported by the Core Research Grant by SERB, Department of Science and Technology, Government of India, EMR/2017/001594. K. Kanagaraj would like to thank National Institute of Technology Karnataka, India, for the financial support.

References

[1] J. I. Al’ber, The solution by the regularization method of operator equations of the first kind with accretive operators in a Banach space, Differ. Uravn. 11 (1975), no. 12, 2242–2248, 2302. Search in Google Scholar

[2] Y. Alber and I. Ryazantseva, Nonlinear Ill-Posed Problems of Monotone Type, Springer, Dordrecht, 2006. Search in Google Scholar

[3] N. Buong, Convergence rates in regularization for nonlinear ill-posed equations under accretive perturbations, Zh. Vychisl. Mat. Mat. Fiz. 44 (2004), no. 3, 397–402. Search in Google Scholar

[4] N. Buong, On nonlinear ill-posed accretive equations, Southeast Asian Bull. Math. 28 (2004), no. 4, 595–600. Search in Google Scholar

[5] N. Buong and N. T. H. Phuong, Convergence rates in regularization for nonlinear ill-posed equations involving m-accretive mappings in Banach spaces, Appl. Math. Sci. (Ruse) 6 (2012), no. 61–64, 3109–3117. Search in Google Scholar

[6] H. Egger and A. Neubauer, Preconditioning Landweber iteration in Hilbert scales, Numer. Math. 101 (2005), no. 4, 643–662. Search in Google Scholar

[7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. Search in Google Scholar

[8] S. George and M. Kunhanandan, An iterative regularization method for ill-posed Hammerstein type operator equation, J. Inverse Ill-Posed Probl. 17 (2009), no. 9, 831–844. Search in Google Scholar

[9] 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. Search in Google Scholar

[10] S. George and M. T. Nair, An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales, Int. J. Math. Math. Sci. (2003), no. 39, 2487–2499. Search in Google Scholar

[11] S. George and M. T. Nair, A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations, J. Complexity 24 (2008), no. 2, 228–240. Search in Google Scholar

[12] S. George and M. T. Nair, A derivative-free iterative method for nonlinear ill-posed equations with monotone operators, J. Inverse Ill-Posed Probl. 25 (2017), no. 5, 543–551. Search in Google Scholar

[13] S. George, S. Pareth and M. Kunhanandan, Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales, Appl. Math. Comput. 219 (2013), no. 24, 11191–11197. Search in Google Scholar

[14] A. Goldenshluger and S. V. Pereverzev, Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations, Probab. Theory Related Fields 118 (2000), no. 2, 169–186. Search in Google Scholar

[15] B. Hofmann, B. Kaltenbacher and E. Resmerita, Lavrentiev’s regularization method in Hilbert spaces revisited, Inverse Probl. Imaging 10 (2016), no. 3, 741–764. Search in Google Scholar

[16] Q.-N. Jin, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse Problems 16 (2000), no. 1, 187–197. Search in Google Scholar

[17] S. G. Kreĭn and J. I. Petunin, Scales of Banach spaces, Russian Math. Surveys 21 (1966), 85–160. Search in Google Scholar

[18] M. T. Nair, Functional Analysis: A First Course, 4th print, Prentice-Hall, New Delhi, 2014. Search in Google Scholar

[19] F. Liu and M. Z. Nashed, Tikhonov regularization of nonlinear ill-posed problems with closed operators in Hilbert scales, J. Inverse Ill-Posed Probl. 5 (1997), no. 4, 363–376. Search in Google Scholar

[20] 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. Search in Google Scholar

[21] P. Mahale, Simplified iterated Lavrentiev regularization for nonlinear ill-posed monotone operator equations, Comput. Methods Appl. Math. 17 (2017), no. 2, 269–285. Search in Google Scholar

[22] P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems 19 (2003), no. 3, 789–803. Search in Google Scholar

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

[24] A. 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. Search in Google Scholar

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

[26] A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales, Numer. Math. 85 (2000), no. 2, 309–328. Search in Google Scholar

[27] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. Search in Google Scholar

[28] E. V. Semenova, Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Comput. Methods Appl. Math. 10 (2010), no. 4, 444–454. Search in Google Scholar

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

[30] 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. Search in Google Scholar

[31] U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), no. 1, 191–207. Search in Google Scholar

[32] V. Vasin and S. George, An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems, Appl. Math. Comput. 230 (2014), 406–413. Search in Google Scholar

Received: 2018-01-19
Revised: 2018-05-18
Accepted: 2018-06-20
Published Online: 2018-07-07
Published in Print: 2019-10-01

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