A simplified homotopy perturbation method for solving nonlinear ill-posed operator equations in Hilbert spaces

Sharad Kumar Dixit

doi:10.33993/jnaat541-1544

Authors

Sharad Kumar Dixit Government Engineering College Kaimur, Bihar Engineering University, DSTTE Patna, Bihar, India

DOI:

https://doi.org/10.33993/jnaat541-1544

Keywords:

Nonlinear ill-posed operator equations, Iterative regularization methods, Nonlinear operators on Hilbert spaces, Convergence analysis, Homotopy perturbation method

Abstract views: 178

Abstract

One popular regularization technique for handling both linear and nonlinear ill-posed problems is homotopy perturbation. In order to solve nonlinear ill-posed problems, we investigate an iteratively-regularized simplified version of the Homotopy perturbation approach in this study. We examine the method's thorough convergence analysis under typical circumstances, focusing on the nonlinearity and the convergence rate under a H\"{o}lder-type source condition. Lastly, numerical simulations are run to confirm the method's effectiveness.

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References

A. B. Bakushinskii, The problem of the iteratively regularized Gauss–Newton method, Comput. Math. Math. Phys., 32 (1992), pp. 1353–1359.

A. B. Bakushinskii, Iterative methods for solving non-linear operator equations without condition of regularity, Soviet. Math. Dokl., 330 (1993), pp. 282–284.

A. Binder, M. Hanke and O. Scherzer, On the Landweber iteration for nonlinear ill-posed problems, J. Inverse Ill-Posed Probl., 4 (1996), no. 5, pp. 381–389. https://doi.org/10.1515/jiip.1996.4.5.381

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse problems, Math. Appl. 375, Kluwer Academic publishers group, Dordrecht, 1996.

H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), no. 4, pp. 523–540.

H. S. Fu, L. Cao, B. Han, A homotopy perturbation method for well log constrained seismic waveform inversion. Chin. J. Geophys.-Chinese Edition, 55 (2012), no. 9, pp. 3173–3179.

M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), no. 1, pp. 21–37.

Q. Jin Q, A general convergence analysis of some Newton-Type methods for nonlinear inverse problems. SIAM J. Numer. Anal., 49 (2011), no. 49, pp. 549–573.

Q. Jin, On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems, Numer. Math., 115 (2010), no. 2, pp. 229–259.

J. Jose and M. P. Rajan, A Simplified Landweber Iteration for Solving Nonlinear Ill-Posed Problems, Int. J. Appl. Comput. Math., 3 (2017), no. 1, pp. 1001–1018.

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, De Gruyter, New York (2008).

B. Kaltenbacher, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems, Inverse Problems, 16 (2000), pp. 1523–1539.

L. Cao, B. Han, W. Wang, Homotopy perturbation method for nonlinear ill-posed operator equations, J. Nonlinear Sci. Numer. Simul., 10 (2009), no. 10, pp. 1319–1322.

L. Cao, B. Han, Convergence analysis of the Homotopy perturbation method for solving nonlinear ill-posed operator equations, Comput. Math. Appl., 61 (2011), no. 8, pp. 2058–2061.

P. Mahale, M. Nair, A simplified generalized Gauss-Newton method for nonlinear ill-posed problems, Math. Comp., 78 (2009), no. 265, pp. 171–184.

P. Mahale, Simplified iterated Lavrentiev regularization for nonlinear ill-posed monotone operator equations, Comput. Methods Appl. Math., 17 (2017), no. 2, pp. 269–285.

P. Mahale and P. K. Dadsena, Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales, Comput. Methods Appl. Math., 18 (2018), no. 4, pp. 687–702.

P. Mahale and K. D. Sharad, Error estimate for simplified iteratively regularized Gauss-Newton method in Banach spaces under Morozove Type stopping rule, J. Inverse Ill-Posed Probl., 20 (2017), no. 2, pp. 321–341.

P. Mahale and K. D. Sharad, Convergence analysis of simplified iteratively regularized Gauss-Newton method in Banach spaces setting, Appl. Anal., 97 (2018), no. 15, pp. 2686–2719.

A. Neubauer, On Landweber iteration for nonlinear ill-posed problems in Hilbert scales, Numer. Math., 85 (2000), pp. 309–328.

O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, Appl. Math. Optim., 38 (1998), pp. 45–68.

O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194 (1995), pp. 911–933.

O. Scherzer, H. W. Engl and K. Kunisch, Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J. Numer. Anal., 30 (1993), pp. 1796–1838.