Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications. Volume II

Ioannis K. Argyros
Cameron University, Department of Mathematical Sciences, Lawton, OK, USA

Santhosh George
Department of Mathematical and Computational Sciences, NIT Karnakata, India

Narayan Thapa
Cameron University, Department of Mathematical Sciences, Lawton, OK, USA

Series: Mathematics Research Developments
BISAC: MAT017000

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This book is dedicated to the approximation of solutions of nonlinear equations using iterative methods. The study about convergence matter of iterative methods is usually based on two categories: semi-local and local convergence analysis. The semi-local convergence category is, based on the information around an initial point, to provide criteria ensuring the convergence of the method; while the local one is, based on the information around a solution, to find estimates of the radii of the convergence balls. The book is divided into two volumes. The chapters in each volume are self-contained so they can be read independently.

Each chapter contains semi-local and local convergence results for single, multi-step and multi-point old and new contemporary iterative methods involving Banach, Hilbert or Euclidean valued operators. These methods are used to generate a sequence defined on the aforementioned spaces that converges with a solution of a nonlinear equation, an inverse problem or an ill-posed problem. It is worth mentioning that most problems in computational and related disciplines can be brought in the form of an equation using mathematical modelling. The solutions of equations can be found in analytical form only in special cases. Hence, it is very important to study the convergence of iterative methods.

The book is a valuable tool for researchers, practitioners, graduate students, and can also be used as a textbook for seminars in all computational and related disciplines.

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)

Chapter 1:

Chapter 2:

Chapter 3:

Chapter 4:

Chapter 5:

Chapter 6:

Chapter 7:

Chapter 8:

Chapter 9:

Chapter 10:

Chapter 11:
[1] Amat, S., Hern´andez, M. A. and Romero, N., A modified Chebyshev’s iterative
method with at least sixth order of convergence, Appl. Math. Comput. 206(1),
(2008), 164-174.
[2] Amat, S., Hern´andez, M.A. and Romero, N., Semilocal convergence of a sixth order
iterativemethod for quadratic equations, Applied NumericalMathematics, 62 (2012),
833-841.
[3] Argyros, I. K., A unifying local semi-local convergence analysis and applications for
two-point Newton-like methods in Banach space, Journal of Mathematical Analysis
and Applications, 298(2)(2004) 374-397,
[4] Argyros, I. K. and Ren, H., On an improved local convergence analysis for the Secant
method, Numer. Algor., 52(2009) 257–271.
[5] Argyros, I. K. and Hilout, S. Weaker conditions for the convergence of Newton’s
method, J. Complexity, 28, (2012), 364-387.
[6] Argyros, I. K., Magre˜n´an, A. A., Iterative methods and their dynamics with applications,
CRC Press, New York, 2017.
[7] Bosarge, W. E. and Falb, P. L., A multipoint method of third order, J. Optimiz. Th.
Appl., 4(1969), 156-166.
[8] Bosarge,W. E. and Falb, P. L., Infinite dimensional multipointmethods and the solution
of two point boundary value problems, Numer. Math., 14(1970), 264–286.
[9] Dennis, J. E., On the Kantorovich hypothesis for Newton’s method, SIAM. J. Numer.
Anal., 6(1969), 493-507.
[10] Dennis, J. E., Toward a unified convergence theory for Newton-likemethods, In Nonlinear
functional analysis and applications, L. B. Rall, Ed., Academic Press, 1971,
New York.
[11] Deuflhard, P. and Heindl, G., Affine invariant convergence theorems for Newton’s
method and extensions to related methods, Siam J. Numer. Anal. 16(1979) 1–10.
[12] Ezquerro, J. A., Hern´andez, M. A. and Rubio,M. J., Secant-like methods for solving
nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math. 115,
1-2(2000) 245–254.
[13] Gragg, W. B. and Tapia, R. A., Optimal error bounds for the Newton-Kantorovich
theorem, SIAM J. Numer. Anal., 11, 1(1974), 10-13.
[14] Hern´andez, M. A. and Rubio, M. J., A uniparameteric family of iterative processes
for solving nondifferentiable equations, J Math. Anal. Appl., 275(2002) 821–834.
[15] Hern´andez-Ver ´on, M. A. and Rubio, M. J., On the ball of convergence of Secantlikemethods
for non-differentiable operators, AppliedMathematics and Computation
(2015), https://doi.org/10.1016/j.amc.2015.10.007.
[16] Hern´andez, M. A., Rubio, M. J. and Ezquerro, J. A., Solving a special case of conservation
problems by Secant-like methods, Appl. Math. Comput., 169, 2(2005)
926–9420.
[17] Kornstaedt,H. J., Ein allgemeiner Konvergenzsatz f ¨ur verach¨arfte Newton-Verfahren
[A general convergence theorem for more stringent Newtonian methods], ISNM 28,
Birkha¨user Verlag, Basel and Stuttgart, 1975, 53-69.
[18] Laasonen, P., Ein ¨uberquadratisch konvergenter iterative algorithmus [An over-square
convergent iterative algorithm], Ann. Acad. Sci. Fenn. Ser I, 450(1969), 1-10.
[19] Magre˜n´an, A. A., Different anomalies in a Jarratt family of iterative root finding
methods, Appl. Math. Comput. 233, (2014), 29-38.
[20] Magre˜n´an, A. A., A new tool to study real dynamics: The convergence plane, Appl.
Math. Comput. 248, (2014), 29-38.
[21] Miel, G. J., The Kantorovich theorem with optimal error bounds, Amer. Math.
Monthly., 86(1979), 212-215.
[22] Miel, G. J., An updated version of the Kantorovich theorem for Newton’s method,
Technical summary report, Mathematical research center, University of Wisconsin,
Madison, 1980.
[23] Moret, I., On a general iterative scheme for Newton-type methods, Numer. Funct.
Anal. and Optimiz, 9(11-12), (1987-1988), 1115-1137.
[24] Ortega, J. M. and Rheinboldt. W.C., Iterative solution of nonlinear equations in
several variables, Academic Press, 1970, New York.
[25] Potra, F. A., On the convergence of a class of Newton-like methods, Iterative solution
of nonlinear systems of equations, Lecture notes in Mathematics 953, Springer
Verlag, 1982, New York.
[26] Potra, F. A. and Pt´ak, V., Nondiscrete Induction and Iterative methods, Pitman Publishing
Limited, London, 1984.
[27] Potra, F. A., Sharp error bounds for a class of Newton-like methods, Libertas Mathematica,
Vol., 5, (1985), 71–83.
[28] Ren, H. and Argyros, I. K., Local convergence of efficient Secant-type methods for
solving nonlinear equations, Appl. Math. Comput., 218(2012) 7655–7664.
[29] Traub, J. F., Iterative methods for the solutions of equations, Prentice-Hall, Englewood
Cliffs, New Jersey, 1964.
[30] Zabrejko, P. P. and Nguen. D. .F, The majorant method in the theory of Newton-
Kantorovich approximations and Pt´ak error estimates, Numer. Funct. Anal. and
Optimiz., 9(1987), 671-684.
[31] Zehnder, E. J., A remark on Newton’s method, Communus Pure Appl. Math, 27
(1974), 361-366.
[32] Yamamoto, T., Error bounds for Newton-like methods under Kantorovich type Assumptions,
MCR Technical Summary Report Nr. 2846, University of Wisconsin,
Madison (1985).

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.

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