Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

Year 2023, Volume: 52 Issue: 3, 729 - 752, 30.05.2023

Chitra Mekoth , Santhosh George , Jidesh P

https://doi.org/10.15672/hujms.1092739

Cited By: 1

https://izlik.org/JA88FD28AG

Abstract

One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh, Appl. Math. Comput. 392, 125701, 2021] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

Keywords

ill-posed problem , fractional Tikhonov regularization , Hilbert scales , parameter choice strategy , finite dimensional realization

References

• [1] S. Djennadi, N. Shawagfeh and O. A. Arqub, A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations, Chaos, Soliton Fractals 150, 111127, 2021.
• [2] S. Djennadi, N. Shawagfeh and O. A. Arqub, A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the timespace fractional diffusion equation, Partial Differential Equations in Applied Mathematics, 4, 100164, 2021.
• [3] H. Egger, Semi-iterative regularization in Hilbert scales, Siam J. Numer. Anal. 44 (1), 66–81, 2006.
• [4] H. Egger and B. Hofmann, Tikhonov regularization in Hilbert scales under conditional stability assumptions, arXiv:1807.05807v1[math.NA], 16 July 2018.
• [5] H. W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Springer Science & Business Media, 1996.
• [6] S. George and M. T. Nair, Error bounds and parameter choice strategies for simplified regularization in Hilbert scales, Inter. Equ. Oper. Theory 29, 231–242, 1997.
• [7] D. Gerth, E. Klann, R. Ramlau and L. Reichel, On fractional Tikhonov regularization, J. Inverse & Ill-Posed Problems, 23 (6), 611–625, 2015.
• [8] C. W. Groetsch, Generalized inverses of linear operators: Representation and Approximation, Marcel Dekker, INC, New York, 1977.
• [9] J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover Publications, New York, 1953.
• [10] M. E. Hochstenbach, S. Noschese and L. Reichel, Fractional regularization matrices for linear discrete ill-posed problems, J. Eng. Math. 93 (1), 113–129, 2015.
• [11] M. E. Hochstenbach and L, Reichel, Fractional Tikhonov regularization for linear discrete ill-posed problems, BIT, 51 (1), 197–215, 2011.
• [12] E. Klann and R. Ramlau, Regularization by fractional filter methods and data smoothing, Inverse Problems, 24 (2), 025018, 2008.
• [13] S. Lu, S. V. Perverzyev, Y. Shao and U. Tautenhahn, On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, J. Integral Equ. Appl. 22 (3), 483-517, 2010.
• [14] P. Mahale and P. K. Dadsene, Simplified generalized Gauss-Newton method for nonlinear ill-posed operator equations in Hilbert scales, Comput. Methods. 18 (4), 687-702, 2018.
• [15] P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (3), 789–803, 2003.
• [16] C. Mekoth, S. George and P. Jidesh, Fractional Tikhonov regularization method in Hilbert scales, Appl. Math. Comput. 392, 125701, 2021.
• [17] C. Mekoth, S. George, P. Jidesh and S. M. Erappa, Finite dimensional realization of fractional Tikhonov regularization method in Hilbert scales, Partial Differential Equations in Applied Mathematics, 5, 100246, 2022.
• [18] S. Mohammady and M. R. Eslahchi, Extension of Tikhonov regularization method using linear fractional programming, J. Comput. Appl. Math. 371, 112677, 2020.
• [19] S. Morigi, L. Reichel and F. Sgallari, Fractional Tikhonov regularization with a nonlinear penalty term, J. Comput. Appl. Math. 324, 142–154, 2017.
• [20] A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales, Appl. Anal. 46 (1-2), 59–72, 1992.
• [21] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9 (1), 84–97, 1962.
• [22] J. Qi-nian, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse Problems, 16 (1), 187–197, 2000.
• [23] M. Rezghi and S. M. Hosseini, A new variant of L-curve for Tikhonov regularization, J. Comput. Appl. Math. 231, 914924, 2009.
• [24] T. Schroter, and U. Tautenhahn, Error estimates for tikhonov regularization in hilbert scales, Numer. Funct. Anal. and Optim. 15, 155168, 1994.
• [25] Jr. C. B. Shaw, Improvements of the resolution of an instrument by numerical solution of an integral equation, J. Math. Anal. Appl. 37, 83–112, 1972.
• [26] Y. Sun, Y. Zhang and Y. Wen, Image Reconstruction Based on Fractional Tikhonov Framework for Planar Array Capacitance Sensor, in IEEE Transactions on Computational Imaging, 8, 109-120, 2022.
• [27] U. Tautenhahn, On a general regularization scheme for non-linear ill-posed problems: II. Regularization in Hilbert scales, Inverse Problems, 14 (6), 1607–1616, 1998.