## Table of Contents

**Table of Contents**

Preface

Chapter 1. Unifying Semilocal and Local Convergence of Newton’s Method (pp. 1-14)

Chapter 2. Higher Order Methods for Nonlinear System of Equations (pp. 15-36)

Chapter 3. Semilocal Convergence of Newton’s Method (pp. 37-46)

Chapter 4. Newton-Kantorovich-Like Theorems under W Conditions (pp. 47-62)

Chapter 5. Unified Local Convergence for Newton-Kantorovich Method under W Condition (pp. 63-74)

Chapter 6. Mesh Independence for Solving Nonlinear Equations (pp. 75-88)

Chapter 7. Expanding the Applicability of Four Iterative Methods (pp. 89-102)

Chapter 8. Improved Complexity of a Homotopic Method for Locating an Approximate Zero (pp. 103-110)

Chapter 9. Convergence Analysis of Frozen Secant-Type Methods (pp. 111-128)

Chapter 10. Unified Convergence Analysis of Frozen Newton-Like Methods (pp. 129-144)

Chapter 11. Solvability of Equations Using Secant-Type Methods (pp. 145-156)

Chapter 12. Newton-Tikhonov Method for Ill-Posed Equations (pp. 157-176)

Chapter 13. Simplified Newton-Tikhonov Regularization Method (pp. 177-188)

Chapter 14. Two Step Newton Lavrentiev Method for Ill-Posed Problems (pp. 189-206)

Chapter 15. Two Step Newton-Tikhonov Methods for Ill-Posed Problems (pp. 207-228)

Chapter 16. Regularization Methods for Ill-Posed Problems (pp. 229-244)

Chapter 17. Expanding the Applicability of Lavrentiev Regularization Method (pp. 245-260)

Chapter 18. Iterated Lavrentiev Regularization (pp. 261-276)

Chapter 19. On The Semilocal Convergence of a Two-Step Newton-Like Projection Method for Ill-Posed Equations (pp. 277-296)

Chapter 20. Local Convergence of Lavrentiev Regularization for Ill-Posed Equations (pp. 297-308)

Chapter 21. Modified Gauss-Newton Method for Nonlinear Ill-Posed Problems (pp. 309-320)

Chapter 22. Two Step Newton-Type Projection Method for Ill-Posed Problems (pp. 321-344)

Chapter 23. Discretized Newtontikhonov Method for Ill-Posed Hammerstein Type Equations (pp. 345-366)

Authors Contact Information (pp. 367-368)

Index (pp. 369)

**References**

Chapter 1:

Chapter 2:

Chapter 3:

Chapter 4:

Chapter 5:

Chapter 6:

Chapter 7:

Chapter 8:

Chapter 9:

Chapter 10:

Chapter 11:

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Chapter 12:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput.(AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] Bakushinsky, A. and Smirnova, A. (2005), On application of generalized discrepancy

principle to iterativemethods for nonlinear ill-posed problems, Numerical Func.

Anal. and Optimization 26, 35-48.

[7] Binder, A., Engl, H.W. and Vessela, S. (1990), Some inverse problems for a nonlinear

parabolic equation connected with continuous casting of steel: stability estimate and

regularization, Numer. Funct. Anal. Optim., 11, 643-671.

[8] Engl. H.W, Hanke. Mand Neubauer. A (1990), Tikhonov regularization of nonlinear

differential equations, Inverse Methods in Action, P.C. Sabatier, ed., Springer-Verlag,

New York, 92-105.

[9] Engl, H. W., Kunisch, K. and Neubauer, A. (1989),Convrgence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems, 5, 523-540.

[10] Groetsch, C. W. (1984), Theory of Tikhonov regularization for Fredholm Equation of

the first kind, Pitmann Books, London.

[11] Groetsch, C.W. and Guacaneme, J. E. (1987), Arcangeli’smethod for Fredhomequations

of the first kind, Proc. Amer. Math. Soc. 99, 256-260.

[12] Guacaneme, J. E. (1990), Aparameter choice for simplified regularization, Rostak,

Math. Kolloq., 42, 59-68.

[13] George, S. (2006), Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Problems, 2, 14, 135-146.

[14] George, S. (2006), Newton-Lavrentiev regularization of ill-posed Hammerstein type

operator equation, J. Inverse and Ill-Posed Problems, 6, 14, 573-582.

[15] George, S. and Elmahdy, A. I. (2012), A quadratic convergence yielding iterative

method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math.

12, no.1, 32-45.

[16] George, S. and Kunhanandan, M. (2009), An iterative regularization method for Illposed

Hammerstein type operator equation, J. Inv. Ill-Posed Problems, 17, 831–844.

[17] George, S. and Kunhanandan, M. (2010), Iterative regularization methods for illposed

Hammerstein type operator equation with monotone nonlinear part, Int. Journal

of Math. Analysis, Vol. 4, no. 34, 1673-1685.

[18] George, S. and Nair. M. T. (1993), An a posteriori parameter choice for simplified

regularization of ill-posed problems, Integr. Equat. Oper. Th. Vol. 16, 392-399.

[19] George, S. and Nair, M. T. (1998), On a generalized Arcangeli’s method for

Tikhonov regularization with inexact data, Numer. Funct. Anal. and Optimiz., 19

(No. 7 and 8), 773-787.

[20] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity, 24,

228-240.

[21] Hanke, M., Neubauer. A. and Scherzer, O. (1995), A convergence analysis of

Landweber iteration of nonlinear ill-posed problems. Numer. Math., 72, 21-37.

[22] Qi-nian, J. and Zong-yi, H. (1996), Finite-dimensional approximations to the solutions

of nonlinear ill-posed problems. Appl. Anal., 62, 253-261.

[23] Qi-nian, J. and Zong-yi, H. (1999), On an a posteriori parameter choice strategy for

Tikhonov regularization of nonlinear ill-posed problems. Numer. Math., 83, 139-159.

[24] Kantorovich. L. V. and Akilov, G. P. (1964), Functional Analysis in Normed Spaces,

Pergamon Press, New York.

[25] Mair, B. A. (1994), Tikhonov regularization for finitely and infinitely smoothing operators,

SIAM J. Math. Anal; 25, 135-147.

[26] Pereverzev, S. and Schock, E. (2005), On the adaptive selection of the parameter in

regularization of ill-posed problems, SIAM. J. Numer. Anal., 43, 5, 2060-2076.

[27] Scherzer, O. (1993), A parameter choice for Tikhonov regularization for solving nonlinear

inverse problems leading to optimal rates. Appl. Math., 38, 479-487.

[28] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM.

J. Numer. Anal, 30, No. 6, 1796-1838.

[29] Scherzer, O. (1992), The use of Tikhonov regularization in the identification of electrical

conductivities from overdetermined boundary data, Results Mathematics, 22,

pp. 599–618.

[30] Scherzer, O., Engl, H.W. and Anderssen, R. S. (1993), Parameter identification from

boundary measurements in parabolic equation arising from geophysics, Nonlinear

Anal., 20, 127-156.

[31] Raus, T. (1984), On the discrepancy principle for the solution of ill-posed problems,

Uch. Zap. Tartu. Gos. Univ., 672, pp. 16–26 (In Russian).

[32] Schock, E. (1984), On the asymptotic order of accuracy of Tikhonov regularization,

J. Optim. Th. and Appl., 44, 95-104.

[33] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 13:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros., I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[3] Argyros, I. K. and Hilout, S. (2012), Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[4] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[5] Bakushinsky, A. and Smirnova, A. (2005), On application of generalized discrepancy

principle to iterativemethods for nonlinear ill-posed problems, Numerical Func.

Anal. and Optimization 26, 35-48.

[6] Binder, A., Engl, H. W., and Vessela, S. (1990), Some inverse problems for a nonlinear

parabolic equation connected with continuous casting of steel: stability estimate

and regularization, Numer. Funct. Anal. Optim., 11, 643-671.

[7] Engl, H. W., Hanke, M. and Neubauer, A. (1990), Tikhonov regularization of nonlinear

differential equations, Inverse Methods in Action, P.C. Sabatier, ed., Springer-

Verlag, New York, 92-105.

[8] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convrgence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems, 5, 523-540.

[9] Groetsch, C. W. (1984), Theory of Tikhonov regularization for Fredholm Equation of

the first kind, Pitmann Books, London.

[10] Groetsch, C.W. and Guacaneme, J. E. (1987), Arcangeli’smethod for Fredhomequations

of the first kind, Proc. Amer. Math. Soc. 99, 256-260.

[11] Guacaneme, J. E. (1990), Aparameter choice for simplified regularization, Rostak,

Math. Kolloq., 42, 59-68.

[12] George, S. (2006), Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Problems, 2, 14, 135-146.

[13] George, S. (2006), Newton-Lavrentiev regularization of ill-posed Hammerstein type

operator equation, J. Inverse and Ill-Posed Problems, 6, 14, 573-582.

[14] George, S. and Elmahdy, A. I. (2012), A quadratic convergence yielding iterative

method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math.

12, no.1, 32-45.

[15] George, S. and Kunhanandan, M. (2009), An iterative regularization method for Illposed

Hammerstein type operator equation, J. Inv. Ill-Posed Problems, 17, 831–844.

[16] George, S. and Kunhanandan, M. (2010), Iterative regularization methods for illposed

Hammerstein type operator equation with monotone nonlinear part, Int. Journal

of Math. Analysis, Vol. 4, no. 34, 1673-1685.

[17] George, S. and Nair, M. T. (1993), An a posteriori parameter choice for simplified

regularization of ill-posed problems, Integr. Equat. Oper. Th. Vol. 16, 392-399.

[18] George, S. and Nair, M. T. (1998), On a generalized Arcangeli’s method for

Tikhonov regularization with inexact data, Numer. Funct. Anal. and Optimiz., 19(No.

7 and 8), 773-787.

[19] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity, 24,

228-240.

[20] Hanke, M., Neubauer, A. and Scherzer, O. (1995), A convergence analysis of

Landweber iteration of nonlinear ill-posed problems. Numer. Math., 72, 21-37.

[21] Qi-nian, J and Zong-yi, H. (1996), Finite-dimensional approximations to the solutions

of nonlinear ill-posed problems. Appl. Anal., 62, 253-261.

[22] Qi-nian, J. and Zong-yi, H. (1999), On an a posteriori parameter choice strategy for

Tikhonov regularization of nonlinear ill-posed problems. Numer. Math., 83, 139-159.

[23] Kantorovich, L. V. and Akilov, G. P. (1964), Functional Analysis in Normed Spaces,

Pergamon Press, New York.

[24] Mair, B. A. (1994), Tikhonov regularization for finitely and infinitely smoothing operators,

SIAM J. Math. Anal; 25, 135-147.

[25] Pereverzev, S. and Schock, E. (2005), On the adaptive selection of the parameter in

regularization of ill-posed problems, SIAM. J. Numer. Anal., 43, 5, 2060-2076.

[26] Scherzer, O. (1993), A parameter choice for Tikhonov regularization for solving nonlinear

inverse problems leading to optimal rates. Appl. Math., 38, 479-487.

[27] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM.

J. Numer. Anal, 30, No.6, 1796-1838.

[28] Scherzer, O. (1989), The use of Tikhonov regularization in the identification of electrical

conductivities from overdetermined problems, Inverse Problems, 5, 227-238.

[29] Scherzer, O., Engl. H. W and Anderssen. R. S (1993), Parameter identification from

boundary measurements in parabolic equation arising from geophysics, Nonlinear

Anal., 20, 127-156.

[30] Raus, T. (1984), On the discrepancy principle for the solution of ill-posed problems,

Acta Comment. Univ. Tartuensis, 672, 16-26.

[31] Schock, E. (1984), On the asymptotic order of accuracy of Tikhonov regularization,

J. Optim. Th. and Appl., 44, 95-104.

[32] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 14:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[3] Argyros, I. K. and Hilout, S. (2010), A convergence analysis for directional two-step

Newton methods, Numer. Algor., 55, 503-528.

[4] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[5] Argyros, I. K. and George, S., Expanding the applicability of a Newton-Lavrentiev

regularization method for ill-posed problems (communicated).

[6] Blaschke, B., Neubauer, A. and Scherzer, O. (1997), On convergence rates for the

iteratively regularized Gauss-Newton method, IMA Journal on Numerical Analysis,

17, 421- 436.

[7] Deuflhard, P., Engl, H. W. and Scherzer, O. (1998), A convergence analysis of iterative

methods for the solution of nonlinear ill-posed problems under affinely invariant

conditions, Inverse problems, 14, 1081-1106.

[8] Engl, H. W., Hanke, M., Neubauer, A. (1990), Tikhonov regularization of nonlinear

differential equations, Inverse Methods in Action, P.C. Sabatier, ed., Springer-Verlag,

New York, 92-105.

[9] Engl, H. W., Hanke, M., Neubauer, A. (1996), Regularization of Inverse Problems,

Kluwer, Dordrecht, 375.

[10] George, S. (2006), Newton-Lavrentiev regularization of ill-posed Hammerstein type

operator equation, J. Inv. Ill-Posed Problems, 14, no.6, 573-582.

[11] George, S. and Elmahdy, A. I. (2012), A quadratic convergence yielding iterative

method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math.

12, 1, 32-45.

[12] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity, 24,

228-240.

[13] George, S. and Pareth, S. (2013), An Application of Newton-type IterativeMethod for

the Approximate Implementation of Lavrentiev Regularization, Journal of Applied

Analysis, 19, 181– 196.

[14] Groetsch, C. W., King, J. T. and Murio, D. (1982), Asymptotic analysis of a finite

element method for Fredholm equations of the first kind, in Treatment of Integral

Equations by Numerical Methods, Eds.: C. T. H. Baker and G. F. Miller, Academic

Press, London, 1-11.

[15] Jaan, J. and Tautenhahn, U. (2003), On Lavrentiev regularization for ill-posed problems

in Hilbert scales, Numer. Funct. Anal. Optim., 24, no. 5-6, 531-555.

[16] Jin, Qi-Nian (2000), Error estimates of some Newton-type methods for solving nonlinear

inverse problems in Hilbert scales, Inverse problems, 16, 187-197.

[17] Jin, Qi-Nian (2000), On the iteratively regularized Gauss-Newtonmethod for solving

nonlinear ill-posed problems, Mathematics of Computation, 69, no. 232, 1603-1623.

[18] Kaltenbacher, B. (1998), A posteriori parameter choice strategies for some Newton

type methods for the regularization of nonlinear ill-posed problems, Numer. Math,

79, 501-528.

[19] Kantorovich, L. V. and Akilov, G. P. (1964), Functional Analysis in Normed Spaces,

Pergamon Press, New York.

[20] Kelley, C. T. (1995), Iterative methods for linear and nonlinear equations, SIAM,

Philadelphia.

[21] Mahale, P. and Nair, M. T. (2003), Iterated Lavrentiev regularization for nonlinear

ill-posed problems, ANZIAM Journal, 51, 191-217.

[22] Mathe, M. and Perverzev, S. V. (2003), Geometry of linear ill-posed problems in

variable Hilbert scales, Inverse Problems, 19, no. 3, 789-803.

[23] Nair,M. T. and Ravishankar, P. (2008), Regularized versions of continuous Newton’s

method and continuous modified Newton’s method under general source conditions,

Numer. Funct.Anal. Optim. 29(9-10), 1140–1165.

[24] Ortega, J. M. and Rheinboldt. W. C. (1970), Iterative solution of nonlinear equations

in general variables, Academic Press, New York and London.

[25] Scherzer, O. (1989), The use of Tikhonov regularization in the identification of electrical

conductivities from overdetermined problems, Inverse Problems, 5, 227-238.

[26] Scherzer, O, Engl, H. W. and Anderssen, R. S. (1993), Parameter identification from

boundary measurements in parabolic equation arising from geophysics, Nonlinear

Anal., 20, 127-156.

[27] Perverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter in

regularization of ill-posed problems, SIAM J. Numer. Anal. 43, 2060-2076.

[28] Ramm, A. G. (2007), Dynamical system method for solving operator equations, Elsevier,

Amsterdam.

[29] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., no.4,

444-454.

[30] Tautanhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 15:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput.(AMS). 80, 327-343.

[3] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[4] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] Argyros, I. K., George, S. and Shobha,M. E. (2016), Cubic convergence order yielding

iterative regularization methods for ill-posed Hammerstein type operator equations,

Rend. Circ. Mat. Palermo, DOI 10.1007/s12215-016-0254-x, 1-21.

[7] Argyros, I. K. and Hilout, S. (2010),A convergence analysis for directional two-step

Newton methods, Numer. Algor., 55, 503-528.

[8] Bakushinskii, A. B. and Kokurin, M. Y. (2004), Iterative Methods for Approximate

Solution of Inverse Problems, Springer, Dordrecht.

[9] Chuan-gang Kang and Guo-qiang He (2009), A mixed Newton-Tikhonovmethod for

nonlinear ill-posed problems, Appl. Math. Mech.-Engl. Ed., DOI: 10.1007/s10483-

009-0608-2, Shanghai University and Springer-Verlag, 30(6), 741-752.

[10] Deuflhard, P., Engl, H. W. and Scherzer, O. (1998), A convergence analysis of iterative

methods for the solution of nonlinear ill-posed problems under affinely invariant

conditions, Inverse Problems, 14(5), 1081-1106.

[11] Engl, H. W. (1993), Regularization methods for the stable solution of inverse problems,

Surveys on Mathematics for Industry, 3, 71 – 143.

[12] Engl, H. W., Hanke, M. and Neubauer, A. (1993), Regularization of Inverse Problems,

Dordrecht: Kluwer.

[13] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convergence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems, 5, 523-540.

[14] George, S. (2006), Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Probl., 2, 14, 135-146.

[15] George, S. and Kunhanandan, M. (2009), An iterative regularization method for Illposed

Hammerstein type operator equation, J. Inv. Ill-Posed Probl., 17, 831-844.

[16] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity, 24,

228-240.

[17] George, S. and Shobha, M. E. (2012), Two-Step Newton-Tikhonov Method for

Hammerstein-Type Equations: Finite-Dimensional Realization, ISRN Applied Mathematics,

vol. 2012, Article ID 783579, 22 pages, doi:10.5402/2012/783579. Vol.1,

no.1, 65-78.

[18] Hanke, M. (1997), A regularization Levenberg-Marquardt scheme with application

to inverse groundwater filtration problems. Inverse Problems, 13(1), 79-95.

[19] Hanke, M., Neubauer, A. and Scherzer, O. (1995), A convergence analysis of the

Landweber iteration for nonlinear ill-posed problems. Numerical Mathematics,

72(11), 21-37.

[20] Jin, Qi-nian (2000), On the iteratively regularized Gauss-Newton method for solving

nonlinear ill-posed problems. Math. Comp. 69(232), 1603-1623.

[21] Jin, Qi-nian (2008), A convergence analysis of the iteratively regularized Gauss-

Newton method under the lipschitz condition, Inverse Problems 24(4), 1-16.

[22] Jin, Qi-nian and Hou Zong-yi (1990), On an a posteriori parameter choice strategy for

Tikhonov regularization of nonlinear ill-posed problems, Numer. Math, 83, 139-159.

[23] Kaltenbacher, B., Neubauer, A. and Scherzer, O. (2008), Iterative regularisation

methods for non-linear ill-posed problems, De Gruyter, Berlin, New York.

[24] Kantorovich, L. V. and Akilov, G. P. (1964), Functional Analysis in Normed Spaces,

Pergamon Press, New York.

[25] Kelley, C. T. (1995) Iterative Methods for Linear and Nonlinear Equations, SIAM,

Philadelphia.

[26] Krasnoselskii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevskii, P. E. (1976),

Integral operators in spaces of summable functions, Translated by T. Ando, Noordhoff

International publishing, Leyden

[27] Mahale, P. and Nair, M. T. (2009), A Simplified generalized Gauss-Newton method

for nonlinear ill-posed problems, Math. Comp., 78,(265), 171-184.

[28] Nair,M. T. and Ravishankar, P. (2008), Regularized versions of continuousNewton’s

method and continuous modified Newton’s method under general source conditions,

Numer. Funct. Anal. Optim., 29(9-10), 1140-1165.

[29] Pereverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter

in regularization of ill-posed problems, SIAM. J. Numer. Anal., 43(5), 2060-2076.

[30] Ramm, A. G. (2005), Inverse Problems, Mathematical and Analytical Techniques

with Applications to Engineering, Springer.

[31] Ramm, A. G., Smirnova, A. B. and Favini, A., (2003), Continuous modified

Newton’s-typemethod for nonlinear operator equations. Ann. Mat. Pura Appl. 182,

37-52.

[32] Semenova, E. V., (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods in Appl. Math.,

4, 444-454.

[33] Tautenhahn, U., (2002), On the method of Lavrentiev regularization for nonlinear

ill-posed problems, Inverse Problems, 18, 191-207.

Chapter 16:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, Approximating solutions of equations using Newton’s method with a modified

Newton’s method iterate as a starting point. Rev. Anal. Numer. Theor. Approx.

36, (2007), 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012), Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho, Y. J. and Hilout, S. ( 2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] Hadamard, J. (1952), Lectures on Cauchy’s Problem in Linear Partial Differential

Equations, Dover Publ., New York.

[7] Engl, H. W., Hanke, M. and Neubauer, A. (1990), Tikhonov regularization of nonlinear

differential equations, Inverse Methods in Action, P.C. Sabatier, ed., Springer-

Verlag, New York, 92-105.

[8] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convrgence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems, 5, 523-540.

[9] Guacaneme, J. E. (1990), A parameter choice for simplified regularization, Rostak,

Math. Kolloq., 42, 59-68.

[10] Groetsch, C. W. (1984), Theory of Tikhonov regularization for Fredholm Equation of

the first kind, Pitmann Books, London.

[11] George, S. (2006), Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Prob., 2, 14, 135-146.

[12] George, S. (2006), Newton-Lavrentiev regularization of ill-posed Hammerstein type

operator equation, J. Inverse and Ill-Posed Probl., 6, 14, 573-582.

[13] George, S. and Kunhanandan,M. (2009), An Iterative Regularization Method for Illposed

Hammerstein Type Operator Equation, J. Inv. Ill-Posed Probl., 17, 831-844.

[14] George, S. and Kunhanandan, M. (2010),Iterative regularization methods for illposed

Hammerstein type operator equation with monotone nonlinear part, Int. Journal

of Math. Analysis, Vol. 4, no. 34, 1673-1685.

[15] George, S. and Nair, M. T. (1998), On a generalized Arcangeli’s method for

Tikhonov regularization with inexact data, Numer. Funct. Anal. and Optim., 19

(No. 7 and 8), 773-787.

[16] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization

for nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity,

24, 228-240.

[16] Kantorovich, L. V. and Akilov, G. P. (1964), Functional Analysis in Normed Spaces,

Pergamon Press, New York.

[18] Nair, M. T. and Ravishankar, P. (2008), Regularized versions of continuous newton’s

method and continuous modified newton’s method under general source conditions,

Numer. Funct. Anal. Optim., 29(9-10), pp. 1140-1165.

[19] Ortega. J.M. and Rheinboldt,W. C. (1970), Iterative solution of nonlinear equations

in general variables, Academic Press, New York and London.

[20] Pereverzev, P. V. and Schock, E. (2005), On the adaptive selection of the parameter

in regularization of ill-posed problems, SIAM. J. Numer. Anal., 43, 5, 2060-2076.

[21] Ramm, A. G., Smirnova, A. B. and Favini, A. (2003), Continuous modified

Newton’s-type method for nonlinear operator equations. Ann. Mat. Pura Appl.,

182, pp.37-52.

[22] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM.

J. Numer. Anal, 30, No.6, 1796-1838.

[23] Schock, E. (1984), On the asymptotic order of accuracy of Tikhonov regularization,

J. Optim. Th. and Appl., 44, 95-104.

[24] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18,191-207.

Chapter 17:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. Approximating solutions of equations using Newton’s method with

a modified Newton’s method iterate as a starting point. Rev. Anal. Numer. Theor.

Approx. 36, (2007), 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012), Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K, Cho, Y. J. and Hilout, S. ( 2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] Bakushinskii,A. and Seminova, A. (2005), On application of generalized discrepancy

principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal.

Optim. 26, 35–48.

[7] Bakushinskii,A. and Seminova, A. (2006), A posteriori stopping rule for regularized

fixed point iterations, Nonlinear Anal. 64, 1255–1261.

[8] Bakushinskii, A. and Seminova, A. (2007), Iterative regularization and generalized

discrepancy principle for monotone operato equations, Numer. Funct. Anal. Optim.

28, 13–25.

[9] Binder, A., Engl, H. W., Groetsch, C. W., Neubauer, A. and Scherzer, O. (1994),

Weakly closed nonlinear operators and parameter identification in parabolic equations

by Tikhonov regularization, Appl. Anal. 55, 215–235.

[10] Engl, H. W., Hanke, M. and Neubauer, A. (1993), Regularization of Inverse Problems,

Dordrecht, Kluwer.

[11] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convergence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems 5, 523–540.

[12] Jin, Q. (2000), On the iteratively regularized Gauss-Newton method for solving nonlinear

ill-posed problems, Math. Comp. 69, 1603–1623.

[13] Jin, Q. and Hou, Z. Y. (1997), On the choice of the regularization parameter for ordinary

and iterated Tikhonov regularization of nonlinear ill-posed problems, Inverse

Problems 13, 815–827.

[14] Jin, Q. and Hou, Z. Y. (1990), On an a posteriori parameter choice strategy for

Tikhonov regularization of nonlinear ill-posed problems, Numer. Math. 83, 139–

159.

[15] Mahale, P. and Nair, M. T. (2009), Iterated Lavrentiev regularization for nonlinear

ill-posed problems, ANZIAM 51, 191–217.

[16] Mahale, P. and Nair, M. T. (2007), General source conditions for nonlinear ill-posed

problems, Numer. Funct. Anal. Optim., 28, 111–126.

[17] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J.

Numer. Anal. 30, 1796–1838.

[18] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math. 4, 444–

454.

[19] Tautenhahn, U. (2004), Lavrentiev regularization of nonlinear ill-posed problems,

Vietnam J. Math. 32, 29–41.

[20] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems 18, 191–207.

[21] Tautenhahn, U. and Jin, Q. (2003), Tikhonov regularization and a posteriori rule for

solving nonlinear ill-posed problems, Inverse Problems 19, 1–21.

Chapter 18:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K., Approximating solutions of equations using Newton’s method with

a modified Newton’s method iterate as a starting point. Rev. Anal. Numer. Theor.

Approx. 36, (2007), 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012), Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho, Y. J. and Hilout, S. ( 2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] Bakushinskii,A. and Seminova, A. (2005), On application of generalized discrepancy

principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal.

Optim. 26, 35–48.

[7] Bakushinskii,A. and Seminova, A. (2006), A posteriori stopping rule for regularized

fixed point iterations, Nonlinear Anal. 64, 1255–1261.

[8] Bakushinskii, A. and Seminova, A. (2007), Iterative regularization and generalized

discrepancy principle for monotone operato equations, Numer. Funct. Anal. Optim.

28, 13–25.

[9] Binder, A., Engl, H. W., Groetsch, C. W., Neubauer, A. and Scherzer, O. (1994),

Weakly closed nonlinear operators and parameter identification in parabolic equations

by Tikhonov regularization, Appl. Anal. 55, 215–235.

[10] Binder, A., Engl, H. W., and Vessela, S. (1990), Some inverse problems for a nonlinear

parabolic equation connected with continuous casting of steel: stability estimate

and regularization, Numer. Funct. Anal. Optim. 11, 643 – 671.

[11] Engl,H.W, Hanke,M. and Neubauer, A. (1990), Tikhonov regularization of nonlinear

differential equations, Inverse Methods in Action, P.C. Sabatier, ed., Springer-Verlag,

New York, 92-105.

[12] Engl, H. W., Hanke, M. and Neubauer, A. (1993), Regularization of Inverse Problems,

Dordrecht: Kluwer.

[13] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convergence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems 5, 523 – 540.

[14] Jin, Q. (2000), On the iteratively regularized Gauss-Newton method for solving nonlinear

ill-posed problems, Math. Comp. 69, 1603 – 1623.

[15] Jin, Q. and Zong-Yi, H. (1997), On the choice of the regularization parameter for ordinary

and iterated Tikhonov regularization of nonlinear ill-posed problems, Inverse

Problems, 13, 815-827.

[16] Jin, Q. and Zong-Yi, H. (1990), On an a posteriori parameter choice strategy for

Tikhonov regularization of nonlinear ill-posed problems, Numer. Math. 83, 139 –

159.

[17] Mahale, P. and Nair, M. T. (2009), Iterated Lavrentiev regularization for nonlinear

ill-posed problems, ANZIAM 51, 191 – 217.

[18] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J.

Numer. Anal. 30 (6), 1796 – 1838.

[19] Scherzer, O. (1989), The use of Tikhonov regularization in the identification of electrical

conductivities from overdetermined problems, Inverse Problems 5, 227 – 238.

[20] Scherzer, O., Engl, H.W. and Anderssen, R. S. (1993), Parameter identification from

boundary measurements in parabolic equation arising from geophysics, Nonlinear

Anal. 20, 127 – 156.

[21] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math. 4, 444–

454.

[22] Tautenhahn, U. (2004), Lavrentiev regularization of nonlinear ill-posed problems,

Vietnam J. Math. 32, 29–41.

[23] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems 18, 191–207.

[24] Tautenhahn, U. and Jin, Q. (2003), Tikhonov regularization and a posteriori rule for

solving nonlinear ill-posed problems, Inverse Problems 19, 1–21.

Chapter 19:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K., Approximating solutions of equations using Newton’s method with

a modified Newton’s method iterate as a starting point. Rev. Anal. Numer. Theor.

Approx. 36, (2007), 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012), Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho, Y. J. and Hilout, S. ( 2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6] George, S. and Elmahdy, A. I. (2010), An analysis of Lavrentiev regularization for

nonlinear ill-posed problems using an iterative regularization method, Int. J. Comput.

Appl. Math., 5(3), 369-381.

[7] George, S. and Elmahdy, A. I. (2010), An iteratively regularized projection method

for nonlinear ill-posed problems, Int. J. Contemp. Math. Sciences, 5, no. 52, 2547-

2565.

[8] George, S. and Elmahdy, A. I. (2012), A quadratic convergence yielding iterative

method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math,

12(1), 3245.

[9 George, S. and Elmahdy, A. I. (2010), An iteratively regularized projection method

with quadratic convergence for nonlinear ill-posed problems, Int. J. of Math. Analysis,

4, no. 45, 2211-2228.

[10] George, S. and Pareth, S. (2012), An application of Newton type iterative method

for Lavrentiev regularization for ill-posed equations: Finite dimensional realization,

IJAM, 42:3, 164-170.

[11] Groetsch, C. W., King, J. T. and Murio, D. (1982), Asymptotic analysis of a finite

element method for Fredholm equations of the first kind, in Treatment of Integral

Equations by Numerical Methods, Eds.: C. T. H. Baker and G. F. Miller, Academic

Press, London, 1-11.

[12] Jaan, J. and Tautenhahn, U. (2003), On Lavrentiev regularization for ill-posed problems

in Hilbert scales, Numer. Funct. Anal. Optim., 24, no. 5-6, 531-555.

[13] Kelley, C. T. (1995), Iterative methods for linear and nonlinear equations, SIAM,

Philadelphia.

[14] Mathe, P. and Perverzev, S. V. (2003), Geometry of linear ill-posed problems in variable

Hilbert scales, Inverse Problems, 19, no. 3, 789-803.

[15] Mahale, P. and Nair, M. T. (2009). Iterated Lavrentiev regularization for nonlinear

ill-posed problems. ANZIAM Journal, 51, 191-217.

[16] Perverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter in

regularization of ill-posed problems, SIAM J. Numer. Anal., 43, 2060-2076.

[17] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., no.4,

444-454.

[18] Tautanhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 20:

[1] Airapetyan, R. G. and Ramm, A. G. (2000), Dynamical systems and discretemethods

for solving nonlinear ill-posed problems, Appl. Math. Reviews 1, G. Anastassiou,

(Eds), 491-536.

[2] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[3] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[4] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[5] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[6] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[7] Bakushinskii,A. and Goncharskii, A. (1994), Ill-posed Problems: Theory and Applications,

Dordrecht, Kluwer.

[8] Binder, A., Engl, H. W., Groetsch, C. W., Neubauer, A. and Scherzer, O. (1994),

Weakly closed nonlinear operators and parameter identification in parabolic equations

by Tikhonov regularization, Appl. Anal., 55, 215-235.

[9] Engl, H. W., Hanke, M. and Neubauer, A. (1990), Tikhonov regularization of nonlinear

differential-algebraic equations, Inverse Methods in Action, P.C. Sabatier, ed.,

Springer-Verlag, New York, 92-105.

[10] Engl, H. W., Hanke, M. and Neubauer, A. (1996), Regularization of Inverse Problems.

Dordrecht: Kluwer.

[11] Janno, J. (2000), Lavrentiev regularization of ill-posed problems containing

nonlinear-to-monotone operators with application to autoconvolution equation, Inverse

Problems, 16, 333-348.

[12] Lin, F. and Nashed, M. Z. (1996), Convergence of regularized solutions of nonlinear

ill-posed problems with monotone operators, Partial Differential Equations and

Applications, P. Marcellini et. al. (Eds.), New York, Dekker, 353-361.

[13] George, S. and Pareth, S. (2012), An Application of Newton Type Iterative Method

for Lavrentiev Regularization for Ill-Posed Equations: Finite Dimensional Realization,

IAENG International Journal of Applied Mathematics, 42:3, IJAM 42 3 07.

[14] Pareth, S. and George, S. (2013), Newton type methods for Lavrentiev regularization

of nonlinear ill-posed operator equations, Journal of Applied Analysis, 19, 181– 196.

[15] Krasnoselskii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevskii, P. E. (1976),

Integral operators in spaces of summable functions, Translated by T. Ando, Noordhoff

International publishing, Leyden.

[16] Scherzer, O (1989), The use of Tikhonov regularization in the identification of electrical

conductivities from overdetermined problems, Inverse Problems, 5, 227-238.

[17] Scherzer, O., Engl, H.W. and Anderssen, R. S. (1993), Parameter identification from

boundary measurements in parabolic equation arising from geophysics, Nonlinear

Anal., 20, 127-156.

[18] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., 4, 444-

454.

[19] Tautenhahn, U. (2004), Lavrentiev regularization of nonlinear ill-posed problems,

Vietnam. J. Math., 32, 29-41.

[20] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 21:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K., Cho. Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[6 Bakushinsky, A, Smirnova, A. (2005), On application of generalized discrepancy

principle to iterative methods for nonlinear ill-posed problems, Numer. Funct. Anal.

Optim., 26, 35-48.

[7] Bakushinskii,A. B. (1992), The problemof convergence of the iteratively regularized

Gauss-Newton method, Comput. Math. Math. Phys., 32, 1353-1359.

[8] Bakushinskii, A. B.(1995), Iterative methods without saturation for solving degenerate

nonlinear operator equations, Dokl. Akad. Nauk, 344, 7-8.

[9] Blaschke, B., Neubauer, A. and Scherzer, O. (1997), On convergence rates for the

iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17, 421-436.

[10] Hadamard, J. (1952), Lectures on Cauchy’s Problem in Linear Partial Differential

Equations, Dover Publ., New York.

[11] Groetsch, C. W., King, J. T. and Murio, D. (1982), Asymptotic analysis of a finite

element method for Fredholm equations of the first kind, In: Treatment of Integral

Equations by Numerical Methods, Eds: C.T.H. Baker and G.F. Miller. Academic

Press, London, 1-11.

[12] Hohage, T. (1997), Logarithmic convergence rate of the iteratively regularized Gauss-

Newton method for an inverse potential and an inverse scattering problem, Inverse

Problems, 13, 1279-1299.

[13] Hohage, T. (2000), Regularization of exponentially ill-posed problems, Numer.

Funct. Anal. Optim., 21, 439-464.

[14] Kaltenbacher, B. (1998), A posteriori parameter choice strategies for some Newtontype

methods for the regularization of nonlinear ill-posed problems, Numer. Math.,

79, 501-528.

[15] Kaltenbacher, B. (2008), A note on logarithmic convergence rates for nonlinear

Tikhonov regularization, J. Inverse Ill-Posed Probl., 16, 79-88.

[16] Langer, S. and Hohage, T. (2007), Convergence analysis of an inexact iteratively

regularized Gauss-Newton method under general source conditions, J. Inverse Ill-

Posed Probl., 15, 19-35.

[17] Engl, H. W., Kunisch, K. and Neubauer, A. (1996), Regularization of Inverse Problems,

Kluwer, Dordrecht.

[18] George, S. (2010), On convergence of regularized modified Newton’s method for

nonlinear ill-posed problems, J. Inverse Ill-Posed Probl., 18, 133-146.

[19] George, S. (2013), Newton type iteration for Tikhonov regularization of nonlinear

ill-posed problems, Journal of Mathematics, vol. 2013, Article ID 439316, 9 pages,

doi:10.1155/2013/439316.

[20] Engl, H. W., Kunisch, K. and Neubauer, A. (1989), Convergence rates for Tikhonov

regularization of nonlinear ill-posed problems, Inverse Problems, 5, 523-540.

[21] Mahale, P. and Nair, M. T. (2009), A simplified generalized Gauss-Newton method

for nonlinear ill-posed problems, Math. Comp., 78, no.265, 171-184.

[22] Pereverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter

in regularization of ill-posed problems, SIAM.J. Numer. Anal., 43, 5, 2060-2076.

[23] Lu, S. and Pereverzev, S. V. (2008), Sparsity reconstruction by the standard Tikhonov

method, RICAM-Report No.17.

[24] Scherzer, O., Engl, H. W. and Kunisch, K. (1993), Optimal a posteriori parameter

choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J.

Numer. Anal., 30, 6, 1796-1838.

[25] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., 4, 444-

454.

Chapter 22:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K. and Hilout, S. (2010), A convergence analysis for directional two-step

Newton methods, Numer. Algor., 55, 503-528.

[6] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[7] Engl, H. W., Hanke, M. and Neubauer, A. (1993), Regularization of Inverse Problems,

Dordrecht: Kluwer.

[8] George, S. (2006),Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Problems, 2, 14, 135-146.

[9] George, S. and Elmahdy, A. I. (2012), A quadratic convergence yielding iterative

method for nonlinear ill-posed operator equations, Comput. Methods Appl. Math.,

12(1), 32-45.

[10] George, S. and Kunhanandan, M. (2009), An iterative regularization method for Illposed

Hammerstein type operator equation, J. Inv. Ill-Posed Problems 17, 831-844.

[11] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity 24,

228-240.

[12] George, S. and Shobha, M. E. (2013), On Improving the Semilocal Convergence of

Newton-Type Iterative method for Ill-posed Hammerstein type operator equations,

IAENG- International Journal of AppliedMathematics, 43:2, IJAM-43-2-03.

[13] George, S. and Shobha, M. E. (2012), Two-Step Newton-Tikhonov Method for

Hammerstein-Type Equations: Finite-Dimensional Realization, ISRN Applied Mathematics,

vol. 2012, Article ID 783579, 22 pages, doi:10.5402/2012/783579.

[14] Shobha,M. E., Argyros, I. K. and George, S. (2014), Newton-type iterative methods

for nonlinear ill-posed Hammerstein-type equations, Applicationes Mathematicae,

41, 107–129.

[15] Kelley, C. T. (1995), Iterative Methods for Linear and Nonlinear Equations, SIAM,

Philadelphia.

[16] Kaltenbacher, B., Neubauer, A. and Scherzer, O. (2008), Iterative regularisation

methods for nolinear ill-posed problems, de Gruyter, Berlin, New York.

[17] Krasnoselskii, M. A., Zabreiko, P. P., Pustylnik, E. I. and Sobolevskii, P. E. (1976),

Integral operators in spaces of summable functions, Translated by T. Ando, Noordhoff

International publishing, Leyden.

[18] Nair, M. T. and Ravishankar, P. (2008), Regularized versions of continuous newton’s

method and continuous modified Newton’s method under general source conditions,

Numer. Funct. Anal. Optim. 29(9-10), 1140-1165.

[19] Mahale, P. and Nair, M. T. (2009), A Simplified generalized Gauss-Newton method

for nonlinear ill-posed problems, Math. Comp., 78, no. 265, 171-184.

[20] Pereverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter

in regularization of ill-posed problems, SIAM. J. Numer. Anal., 43, 5, 2060-2076.

[21] Ramm, A. G., Smirnova, A. B. and Favini, A. (2003), Continuous modified

Newton’s-typemethod for nonlinear operator equations. Ann. Mat. Pura Appl. 182,

37-52.

[22] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., no.4,

444-454.

[23] Groetsch, C. W. and Neubauer, A. (1988), Convergence of a general projection

method for an operator equation of the first kind, Houstan. J. Math., 14, 201-208.

[24] Krisch, A. (1996), An introduction to the Mathematical Theory of inverse problems,

Springer, New York.

[25] Perverzev, S. V. and Probdorf, S. (2000), On the characterization of selfregularization

properties of a fully discrete projection method for Symms integral

equation, J. Integral Equat. Appl., 12, 113-130.

[26] Mahale, P. and Nair, M. T. (2009), Iterated Lavrentiev regularization for nonlinear

ill-posed problems, ANZIAM, 51,191-217.

[27] Tautenhahn, U. (2002), On the method of Lavrentiev regularization for nonlinear illposed

problems, Inverse Problems, 18, 191-207.

Chapter 23:

[1] Argyros, I. K. (2008), Convergence and Application of Newton-type Iterations,

Springer.

[2] Argyros, I. K. (2007), Approximating solutions of equations using Newton’s method

with a modified Newton’s method iterate as a starting point. Rev. Anal. Numer.

Theor. Approx. 36, 123-138.

[3] Argyros, I. K. (2011), A Semilocal convergence for directional Newton methods,

Math. Comput. (AMS). 80, 327-343.

[4] Argyros, I. K. and Hilout, S. (2012),Weaker conditions for the convergence of Newton’s

method, J. Complexity, 28, 364-387.

[5] Argyros, I. K. and Hilout, S. (2010), A convergence analysis for directional two-step

Newton methods, Numer. Algor., 55, 503-528.

[6] Argyros, I. K., Cho, Y. J. and Hilout, S. (2012), Numerical methods for equations

and its applications, CRC Press, Taylor and Francis, New York.

[7] Engl, H. W., Hanke, M. and Neubauer, A. (1993), Regularization of Inverse Problems,

Dordrecht: Kluwer.

[8] Engl, H. W. (1993), Regularization methods for the stable solution of inverse problems,

Surveys on Mathematics for Industry, 3, 71-143.

[9] George, S. (2006), Newton-Tikhonov regularization of ill-posed Hammerstein operator

equation, J. Inverse and Ill-Posed Problems, 2, 14, 135-146.

[10] George, S. and Kunhanandan, M. (2009), An iterative regularization method for Illposed

Hammerstein type operator equation, J. Inv. Ill-Posed Problems 17, 831-844.

[11] George, S. and Nair, M. T. (2008), A modified Newton-Lavrentiev regularization for

nonlinear ill-posed Hammerstein-Type operator equation, Journal of Complexity 24,

228-240.

[12] Shobha, M. E., Argyros, I. K. and George. S. (2014), Newton-type iterative methods

for nonlinear ill-posed Hammerstein-type equations, Applicationes Mathematicae,

41, 107–129.

[12] George, S. and Shobha, M. E (2012), A Two-Step Newton-Tikhonov Method for

Hammerstein-Type Equations: Finite-Dimensional Realization, ISRN Applied Mathematics,

vol. 2012, Article ID 783579, 22 pages, doi:10.5402/2012/783579.

[13] Kaltenbacher, B., Neubauer, A. and Scherzer. O. (2008), Iterative regularisation

methods for nolinear ill-posed porblems, de Gruyter, Berlin, New York.

[14] Kelley. C. T (1995), Iterative Methods for Linear and Nonlinear Equations, SIAM,

Philadelphia.

[15] Krisch. A (1996), An introduction to the Mathematical Theory of inverse problems,

Springer, New York.

[16] Nair, M. T. and Ravishankar, P. (2008), Regularized versions of continuous newton’s

method and continuous modified Newton’s method under general source conditions,

Numer. Funct. Anal. Optim. 29(9-10), 1140-1165.

[17] Pereverzev, S. V. and Schock, E. (2005), On the adaptive selection of the parameter

in regularization of ill-posed problems, SIAM. J. Numer. Anal., 43, 5, 2060-2076.

[18] Perverzev, S. V. and Probdorf, S. (2000), On the characterization of selfregularization

properties of a fully discrete projection method for Symms integral

equation, J. Integral Equat. Appl., 12, 113-130.

[19] Ramm, A. G., Smirnova. A. B. and Favini, A. (2003), Continuous modified

Newton’s-typemethod for nonlinear operator equations. Ann. Mat. Pura Appl. 182,

37-52.

[20] Semenova, E. V. (2010), Lavrentiev regularization and balancing principle for solving

ill-posed problems with monotone operators, Comput. Methods Appl. Math., no.4,

444-454.

[21] Shobha,M. E. and George, S. (2012), Dynamical SystemMethod for ill-posed Hammerstein

type operator equations with Monotone Operators, International Journal of

Pure and Applied Mathematics, 81(1), 129-143.