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Research Article Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales Monnanda Erappa Shobha and Santhosh George Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore 575 025, India Correspondence should be addressed to Monnanda Erappa Shobha; [email protected] Received 13 February 2014; Accepted 11 June 2014; Published 1 July 2014 Academic Editor: Beny Neta Copyright © 2014 M. E. Shobha and S. George. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equation () = . In order to improve the error estimate available by Vasin and George (2013), in the present paper we extend the iterative method considered by Vasin and George (2013), in the setting of Hilbert scales. e error estimates obtained under a general source condition on 0 ( 0 is the initial guess and is the actual solution), using the adaptive scheme proposed by Pereverzev and Schock (2005), are of optimal order. e algorithm is applied to numerical solution of an integral equation in Numerical Example section. 1. Introduction In this study, we are interested in approximately solving a nonlinear ill-posed operator equation: () = , (1) where : () ⊆ is a nonlinear operator. Here () is the domain of , and ⟨⋅, ⋅⟩ is the inner product with corresponding norm ‖⋅‖ on the Hilbert spaces and . roughout this paper we denote by ( 0 ) the ball of radius centered at 0 , () denotes the Fr´ echet derivative of at ∈ (), and (⋅) denotes the adjoint of (⋅). We assume that are the available noisy data satisfying ≤ , (2) where is the noise level. Equation (1) is, in general, ill-posed, in the sense that a unique solution that depends continuously on the data does not exist. Since the available data is , one has to solve (approximately) the perturbed equation () = (3) instead of (1). To solve the ill-posed operator equations, various reg- ularization methods are used, for example, Tikhonov reg- ularization, Landweber iterative regularization, Levenberg- Marquardt method, Lavrentiev regularization, Newton type iterative method, and so forth (see, e.g., [116]). In [16], Vasin and George considered the iteration (which is a modified form of the method considered in [8]) +1, := , −1 [ 0 ( ( , )− ) + ( , 0 )] , (4) where 0 := 0 0 + , 0 = ( 0 ), 0, = 0 is the initial guess, >0 is the regularization parameter, and >. Iteration (4) was used to obtain an approximation for the zero of ( 0 ) (( 0 )− ) + ( − 0 )=0 and proved that is an approximate solution of (1). e regularization parameter in [16] was chosen appropriately from the finite set := { :0< 0 < 1 < ⋅⋅⋅ < } depending on the inexact data and the error level satisfying (2) using the adaptive parameter selection procedure suggested by Pereverzev and Schock [17]. In order to improve the rate of convergence many authors have considered the Hilbert scale variant of the regularization Hindawi Publishing Corporation Journal of Mathematics Volume 2014, Article ID 965097, 9 pages http://dx.doi.org/10.1155/2014/965097

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Page 1: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Research ArticleNewton Type Iteration for Tikhonov Regularization ofNonlinear Ill-Posed Problems in Hilbert Scales

Monnanda Erappa Shobha and Santhosh George

Department of Mathematical and Computational Sciences National Institute of Technology Karnataka Mangalore 575 025 India

Correspondence should be addressed to Monnanda Erappa Shobha shobhamegmailcom

Received 13 February 2014 Accepted 11 June 2014 Published 1 July 2014

Academic Editor Beny Neta

Copyright copy 2014 M E Shobha and S George This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Recently Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equation119865(119909) = 119910 In order to improve the error estimate available by Vasin and George (2013) in the present paper we extend the iterativemethod considered by Vasin and George (2013) in the setting of Hilbert scalesThe error estimates obtained under a general sourcecondition on 119909

0minus 119909 (119909

0is the initial guess and 119909 is the actual solution) using the adaptive scheme proposed by Pereverzev and

Schock (2005) are of optimal order The algorithm is applied to numerical solution of an integral equation in Numerical Examplesection

1 Introduction

In this study we are interested in approximately solving anonlinear ill-posed operator equation

119865 (119909) = 119910 (1)

where 119865 119863(119865) sube 119883 rarr 119884 is a nonlinear operator Here119863(119865) is the domain of 119865 and ⟨sdot sdot⟩ is the inner product withcorresponding norm sdot on the Hilbert spaces 119883 and 119884Throughout this paper we denote by 119861

119903(1199090) the ball of radius

119903 centered at 1199090 1198651015840(119909) denotes the Frechet derivative of 119865 at

119909 isin 119863(119865) and 1198651015840lowast(sdot) denotes the adjoint of 1198651015840(sdot) We assumethat 119910120575 isin 119884 are the available noisy data satisfying

10038171003817100381710038171003817119910 minus 11991012057510038171003817100381710038171003817le 120575 (2)

where 120575 is the noise level Equation (1) is in general ill-posedin the sense that a unique solution that depends continuouslyon the data does not exist Since the available data is 119910120575 onehas to solve (approximately) the perturbed equation

119865 (119909) = 119910120575 (3)

instead of (1)

To solve the ill-posed operator equations various reg-ularization methods are used for example Tikhonov reg-ularization Landweber iterative regularization Levenberg-Marquardt method Lavrentiev regularization Newton typeiterative method and so forth (see eg [1ndash16])

In [16] Vasin and George considered the iteration (whichis a modified form of the method considered in [8])

119909120575

119899+1120572= 119909119899120572minus 119877minus1

120572[119860lowast

0(119865 (119909120575

119899120572) minus 119910120575) + 120572 (119909

120575

119899120572minus 1199090)]

(4)

where 1198770= 119860lowast

01198600+ 120573119868 119860

0= 1198651015840(1199090) 119909120575

0120572= 1199090is the initial

guess 120572 gt 0 is the regularization parameter and 120573 gt 120572Iteration (4) was used to obtain an approximation for thezero 119909120575

120572of 1198651015840(119909

0)lowast(119865(1199090) minus 119910120575) + 120572(119909 minus 119909

0) = 0 and proved

that 119909120575120572is an approximate solution of (1) The regularization

parameter 120572 in [16] was chosen appropriately from the finiteset 119863119873= 120572

119894 0 lt 120572

0lt 1205721lt sdot sdot sdot lt 120572

119873 depending

on the inexact data 119910120575 and the error level 120575 satisfying (2)using the adaptive parameter selection procedure suggestedby Pereverzev and Schock [17]

In order to improve the rate of convergencemany authorshave considered theHilbert scale variant of the regularization

Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 965097 9 pageshttpdxdoiorg1011552014965097

2 Journal of Mathematics

methods for solving ill-posed operator equations for exam-ple [18ndash26] In this study we present the Hilbert scale variantof (4)

We consider the Hilbert scale 119883119905119905isinR (see [14 18 23 26ndash

29]) generated by a strictly positive self-adjoint operator 119861 119863(119861) sub 119883 rarr 119883 with the domain119863(119861) dense in119883 satisfying119861119909 ge 119909 for all 119909 isin 119863(119861) Recall [19 28] that the space119883119905is the completion of 119863 = ⋂infin

119896=0119863(119861119896) with respect to the

norm 119909119905 induced by the inner product

⟨119906 V⟩119905 = ⟨1198611199052119906 1198611199052V⟩ 119906 V isin 119863 (119861) (5)

In this paper we consider the sequence 119909120575119899120572119904 defined

iteratively by

119909120575

119899+1120572119904

= 119909120575

119899120572119904minus 119877minus1

120573[119860lowast

0(119865 (119909120575

119899120572119904) minus 119910120575) + 120572119861

119904(119909120575

119899120572119904minus 1199090)]

(6)

where 119877minus1120573= (119860lowast

01198600+120573119861119904)minus1 1199091205750120572119904

= 1199090 is the initial guess

120573 gt 120572 for obtaining an approximation for zero 119909120575120572119904

of (cf[21 30])

119860lowast

0(119865 (119909) minus 119910

120575) + 120572119861

119904(119909 minus 119909

0) = 0 (7)

As in [16] we use the following center-type Lipschitzcondition for the convergence of the iterative scheme

Assumption 1 Let 1199090isin 119883 be fixed There exists a constant 119896

0

such that for every 119906 isin 119861119903(1199090) cup 119861119903(119909) sube 119863(119865) and V isin 119883

there exists an element Φ(1199090 119906 V) isin 119883 satisfying

[1198651015840(1199090) minus 1198651015840(119906)] V = 1198651015840 (1199090)Φ (1199090 119906 V)

1003817100381710038171003817Φ (1199090 119906 V)1003817100381710038171003817 le 1198960 V

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

(8)

The error estimates in this work are obtained using thesource condition on 119909

0minus 119909 In addition to the advantages

listed in [16 see page 3] the method considered in thispaper gives optimal order for a range of values of smoothnessassumptions on 119909

0minus 119909 The regularization parameter 120572 is

chosen from some finite set 1205720lt 1205721lt 1205722sdot sdot sdot lt 120572

119873 using

the balancing principle considered by Pereverzev and Schockin [17]

The paper is organized as follows In Section 2 we givethe analysis of the method for regularization of (6) in thesetting of Hilbert scales The error analysis and adaptivescheme of parameter 120572 are given in Section 3 In Section 4implementation of the method along with a numericalexample is presented to validate the efficiency of the proposedmethod and we conclude the paper in Section 5

2 The Method

First we will prove that the sequence (119909120575119899120572119904) defined by (6)

converges to the zero 119909120575120572119904

of (7) and then we show that 119909120575120572119904

isan approximation to the solution 119909 of (1)

Let 119860119904= 1198600119861minus1199042 We make use of the relation

10038171003817100381710038171003817(119860lowast

119904119860119904+ 120572119868)minus1(119860lowast

119904119860119904)11990110038171003817100381710038171003817le 120572119901minus1 119901 gt 0 0 lt 119901 le 1 (9)

which follows from the spectral properties of the positive self-adjoint operator 119860lowast

119904119860119904 119904 gt 0 Usually for the analysis of

regularization methods in Hilbert scales an assumption ofthe form (cf [18 24])

100381710038171003817100381710038171198651015840(119909) 119909

10038171003817100381710038171003817sim 119909minus119887 119909 isin 119883 (10)

on the degree of ill-posedness is used In this paper instead of(10) we require only a weaker assumption

1198891119909minus119887 le

100381710038171003817100381711986001199091003817100381710038171003817 le 1198892119909minus119887 119909 isin 119863 (119865) (11)

for some positive reals 119887 1198891 and 119889

2

Note that (11) is simpler than that of (10) Now we define119891 and 119892 by

119891 (119905) = min 1198891199051 119889119905

2 119892 (119905) = max 119889119905

1 119889119905

2

119905 isin R |119905| le 1(12)

The following proposition is used for further analysis

Proposition 2 (cf see [29 Proposition 21]) For 119904 gt 0 and|]| le 1

119891 (]) 119909minus](119904+119887) le100381710038171003817100381710038171003817(119860lowast

119904119860119904)]2119909100381710038171003817100381710038171003817le 119892 (]) 119909minus](119904+119887) 119909 isin 119867

(13)

Let us define a few parameters essential for the conver-gence analysis Let

1205952(119904) =

119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887)) 120595

2(119904) =

1

119891 (119904 (119904 + 119887))

119890120575

119899120572119904=10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 forall119899 ge 0

1205750lt1205731198872(119904+119887)

211989601205952(119904)

(1205952 (119904) +

1205722

0

21205732)

1003817100381710038171003817119909 minus 11990901003817100381710038171003817 le 120588

(14)

with

120588 ltminus1

1198960

+1

11989601205952 (119904)

times radic1205952(119904) [(

1205722

21205732+ 1205952(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

120574120588= 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

1198960

21205882+ 120588)

(15)

Journal of Mathematics 3

Further let 120574120588lt 1205722

0411989601205732 and

1199031=

120572 + radic1205722 minus 4119896

01205741205881205732

21198960120573

1199032= min

1

1205952(119904) 1198960

120572 minus radic1205722 minus 4119896

01205741205881205732

21198960120573

(16)

For 119903 isin (1199031 1199032) let

119902 = 1205952(119904) (119896

0119903 +120573 minus 120572

120573) (17)

Then 119902 lt 1

Lemma 3 Let Proposition 2 hold Then for all ℎ isin 119883 thefollowing hold

(a) (119860lowast01198600+ 120573119861119904)minus1119860lowast

01198600ℎ le 120595

2(119904)ℎ

(b) (119860lowast01198600+ 120573119861119904)minus1119861119904ℎ le 120595

2(119904)(1120573)ℎ

Proof Observe that by Proposition 2

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198601199041198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus1(119860lowast

119904119860119904) 1198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904119860119904

10038171003817100381710038171003817

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (s (119904 + 119887))100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))ℎ

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119861119904ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus110038171003817100381710038171003817

100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573

100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573ℎ

(18)

This completes the proof of the lemma

Theorem 4 Let 119890120575119899120572119904

and 119902 be as in (14) and (17) respectivelylet 119909120575119899120572119904

be as defined in (6) with 120575 isin (0 1205750] Then under

Assumption 1 and Lemma 3 the following estimates hold forall 119899 ge 0

(a) 119909120575119899+1120572119904

minus 119909120575

119899120572119904 le 119902119899120574120588

(b) 119909120575119899120572119904

isin 119861119903(1199090)

Proof If 119909120575119899120572119904

isin 119861119903(1199090) then by Assumption 1

119909120575

119899+1120572119904minus 119909120575

119899120572119904

= 119909120575

119899120572119904minus 119909120575

119899minus1120572119904minus (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ 120572119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904) ]

= (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

01198600(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

minus 119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ (120573 minus 120572) 119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)]

= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

+ (119860lowast

01198600+ 120573119861119904)minus1(120573 minus 120572) 119861

119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

= Γ1+ Γ2

(19)

where

Γ1= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

(20)

4 Journal of Mathematics

and Γ2= (119860lowast

01198600+120573119861119904)minus1(120573minus120572)119861

119904(119909120575

119899120572119904minus119909120575

119899minus1120572119904) and hence

by Assumption 1 and Lemma 3(a) we have

1003817100381710038171003817Γ11003817100381710038171003817 =10038171003817100381710038171003817minus(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600

times int

1

0

Φ(119909120575

119899minus1120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904)

1199090 119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 11988911990510038171003817100381710038171003817

le 1205952(119904) 119896011990310038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

(21)

and by Lemma 3(b)

1003817100381710038171003817Γ21003817100381710038171003817 le

120573 minus 120572

1205731205952(119904)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 (22)

Hence by (19) (21) and (22) we have

10038171003817100381710038171003817119909120575

119899+1120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le 1205952(119904) (119896

0119903 +120573 minus 120572

120573)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

= 11990210038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

le 119902119899 10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817= 119902119899119890120575

1120572119904

(23)

Next we show that 1198901205751120572119904

lt 120574120588 using Assumption 1 and

Lemma 3 Observe that

119890120575

1120572119904=10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817

=10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

0(119865 (119909120575

0120572119904) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119865 (1199090) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times [119910120575minus 119910 + 119865 (119909)

minus119865 (1199090) minus 1198600(119909 minus 119909

0) + 1198600(119909 minus 119909

0)]10038171003817100381710038171003817

le10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119910120575minus 119910)

10038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times int

1

0

(1198651015840(1199090+ 119905 (119909 minus 119909

0)) minus 119860

0)

times (119909 minus 1199090) 11988911990510038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198600(119909 minus 119909

0)10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))120573minus1198872(119904+119887)

120575 +119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

11989601205882

2

+119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))120588

le 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

11989601205882

2+ 120588)

= 120574120588lt 119903

(24)

Hence (a) follows from (23) and (24)

To prove (b) note that 1199091205751120572119904

minus 119909120575

0120572119904 le 120574120588lt 119903 Suppose

119909120575

119898120572119904isin 119861119903(1199090) for some119898 then

10038171003817100381710038171003817119909120575

119898+1120572119904minus 1199090

10038171003817100381710038171003817

le10038171003817100381710038171003817119909120575

119898+1120572minus 119909120575

119898120572119904

10038171003817100381710038171003817+ sdot sdot sdot +

10038171003817100381710038171003817119909ℎ120575

1120572119904minus 1199090

10038171003817100381710038171003817

le (119902119898+ 119902(119898minus1)

+ sdot sdot sdot + 1) 119890120575

1120572119904

le1

1 minus 119902119890120575

1120572119904

le

120574120588

1 minus 119902

lt 119903

(25)

Thus by induction 119909120575119899120572119904

isin 119861119903(1199090) for all 119899 ge 0 This proves

(b)Next we go to the main result of this section

Theorem5 Let119909120575119899120572s be as in (6) 120575 isin (0 1205750] and assumptions

ofTheorem 4 holdThen (119909120575119899120572119904) is a Cauchy sequence in 119861

119903(1199090)

and converges say to 119909120575120572119904isin 119861119903(1199090) Further119860lowast

0(119865(119909120575

120572119904)minus119910120575)+

120572119861119904(119909120575

120572119904minus 1199090) = 0 and

10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

120572119904

10038171003817100381710038171003817le 119862119902119899 (26)

where 119862 = 120574120588(1 minus 119902)

Proof Using relation (a) of Theorem 4 we obtain

10038171003817100381710038171003817119909120575

119899+119898120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

10038171003817100381710038171003817119909120575

119899+119894+1120572119904minus 119909120575

119899+119894120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

119902(119899+119894)119890120575

1120572119904

Journal of Mathematics 5

= 119902119899119890120575

1120572119904+ 119902(119899+1)

119890120575

1120572119904

+ sdot sdot sdot + 119902(119899+119898)

119890120575

1120572119904

le 119902119899(1 + 119902 + 119902

2+ sdot sdot sdot + 119902

119898) 119890120575

1120572119904

le 119902119899(1

1 minus 119902) 120574120588

le 119862119902119899

(27)

Thus 119909120575119899120572119904

is a Cauchy sequence in 119861119903(1199090) and hence it

converges say to 119909120575120572119904isin 119861119903(1199090)

Now letting 119899 rarr infin in (6) we obtain

119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 (28)

This completes the proof

The following assumption on source function and sourcecondition is required to obtain the error estimates

Assumption 6 There exists a continuous strictly monotoni-cally increasing function 120593 (0 119860

1199042] rarr (0infin) such that

the following conditions hold

(i) lim120582rarr0

120593(120582) = 0

(ii) sup120582gt0(120572120593(120582)(120582 + 120572)) le 120593(120572) for all 120582 isin (0 119860

1199042]

and

(iii) there exists 119908 isin 119883 with 119908 le 1198642 such that

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus 119909) = 120593 (119860

lowast

119904119860119904) 119908 (29)

Remark 7 If 1199090minus 119909 isin 119883

119905 for example 119909

0minus 119909119905le 1198641 for

some positive constant 1198641and 0 le 119905 le 2119904 + 119887 then we have

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus119909) = 120593(119860

lowast

119904119860119904)119908 where120593(120582) = 120582119905(119904+119887)

119908 = (119860lowast

119904119860119904)(119904minus119905)2(119904+119887)

1198611199042(119909 minus 119909

0) and 119908 le 119892((119904 minus 119905)(119904 +

119887))1198641= 1198642

Theorem 8 Let 119909120575120572119904

be the solution of (7) and supposeAssumptions 1 and 6 hold then

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952 (119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952 (119904) 1198960119903

(30)

Proof Let119872 = int1

01198651015840(119909 + 119905(119909

120575

120572119904minus 119909))119889119905 Then

119865 (119909120575

120572119904) minus 119865 (119909) = 119872(119909

120575

120572119904minus 119909) (31)

Since 119860lowast0(119865(119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 one can see that

(119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

= (119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

minus 119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) minus 120572119861

119904(119909120575

120572119904minus 1199090)

= 119860lowast

0[1198600minus119872] (119909

120575

120572119904minus 119909) + 119860

lowast

0(119910120575minus 119910)

+ 120572119861119904(1199090minus 119909)

119909120575

120572119904minus 119909 = (119860

lowast

01198600+ 120572119861119904)minus1

times [119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

+119860lowast

0(119910120575minus 119910) + 120572119861

119904(1199090minus 119909)]

= 1199041+ 1199042+ 1199043

(32)

where

1199041= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

1199042= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(119910120575minus 119910)

1199043= (119860lowast

01198600+ 120572119861119904)minus1120572119861119904(1199090minus 119909)

(33)

Note that by Assumption 1 and Lemma 3100381710038171003817100381711990411003817100381710038171003817 le 1205952 (119904) 1198960119903

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817 (34)

by Proposition 2100381710038171003817100381711990421003817100381710038171003817 le 1205952(119904)120572

minus1198872(119904+119887)120575 (35)

and by Assumption 6100381710038171003817100381711990431003817100381710038171003817 le 1205952(119904)120593 (120572)

(36)

Hence by (34)ndash(36) and (32) we have

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(37)

This completes the proof of the theorem

21 Error Bounds under Source Conditions

Theorem 9 Let 119909120575119899120572119904

be as in (6) If assumptions inTheorems5 and 8 hold then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119902119899+

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(38)

where 119862 is as in Theorem 5 Further if 119899120575= min119899 119902119899 le

120572minus1198872(119904+119887)

120575 then10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119904(120593 (120572) + 120572

minus1198872(119904+119887)120575) (39)

where 119862119904= 119862 + (120595

2(119904)(1 minus 120595

2(119904)1198960119903))

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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2 Journal of Mathematics

methods for solving ill-posed operator equations for exam-ple [18ndash26] In this study we present the Hilbert scale variantof (4)

We consider the Hilbert scale 119883119905119905isinR (see [14 18 23 26ndash

29]) generated by a strictly positive self-adjoint operator 119861 119863(119861) sub 119883 rarr 119883 with the domain119863(119861) dense in119883 satisfying119861119909 ge 119909 for all 119909 isin 119863(119861) Recall [19 28] that the space119883119905is the completion of 119863 = ⋂infin

119896=0119863(119861119896) with respect to the

norm 119909119905 induced by the inner product

⟨119906 V⟩119905 = ⟨1198611199052119906 1198611199052V⟩ 119906 V isin 119863 (119861) (5)

In this paper we consider the sequence 119909120575119899120572119904 defined

iteratively by

119909120575

119899+1120572119904

= 119909120575

119899120572119904minus 119877minus1

120573[119860lowast

0(119865 (119909120575

119899120572119904) minus 119910120575) + 120572119861

119904(119909120575

119899120572119904minus 1199090)]

(6)

where 119877minus1120573= (119860lowast

01198600+120573119861119904)minus1 1199091205750120572119904

= 1199090 is the initial guess

120573 gt 120572 for obtaining an approximation for zero 119909120575120572119904

of (cf[21 30])

119860lowast

0(119865 (119909) minus 119910

120575) + 120572119861

119904(119909 minus 119909

0) = 0 (7)

As in [16] we use the following center-type Lipschitzcondition for the convergence of the iterative scheme

Assumption 1 Let 1199090isin 119883 be fixed There exists a constant 119896

0

such that for every 119906 isin 119861119903(1199090) cup 119861119903(119909) sube 119863(119865) and V isin 119883

there exists an element Φ(1199090 119906 V) isin 119883 satisfying

[1198651015840(1199090) minus 1198651015840(119906)] V = 1198651015840 (1199090)Φ (1199090 119906 V)

1003817100381710038171003817Φ (1199090 119906 V)1003817100381710038171003817 le 1198960 V

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

(8)

The error estimates in this work are obtained using thesource condition on 119909

0minus 119909 In addition to the advantages

listed in [16 see page 3] the method considered in thispaper gives optimal order for a range of values of smoothnessassumptions on 119909

0minus 119909 The regularization parameter 120572 is

chosen from some finite set 1205720lt 1205721lt 1205722sdot sdot sdot lt 120572

119873 using

the balancing principle considered by Pereverzev and Schockin [17]

The paper is organized as follows In Section 2 we givethe analysis of the method for regularization of (6) in thesetting of Hilbert scales The error analysis and adaptivescheme of parameter 120572 are given in Section 3 In Section 4implementation of the method along with a numericalexample is presented to validate the efficiency of the proposedmethod and we conclude the paper in Section 5

2 The Method

First we will prove that the sequence (119909120575119899120572119904) defined by (6)

converges to the zero 119909120575120572119904

of (7) and then we show that 119909120575120572119904

isan approximation to the solution 119909 of (1)

Let 119860119904= 1198600119861minus1199042 We make use of the relation

10038171003817100381710038171003817(119860lowast

119904119860119904+ 120572119868)minus1(119860lowast

119904119860119904)11990110038171003817100381710038171003817le 120572119901minus1 119901 gt 0 0 lt 119901 le 1 (9)

which follows from the spectral properties of the positive self-adjoint operator 119860lowast

119904119860119904 119904 gt 0 Usually for the analysis of

regularization methods in Hilbert scales an assumption ofthe form (cf [18 24])

100381710038171003817100381710038171198651015840(119909) 119909

10038171003817100381710038171003817sim 119909minus119887 119909 isin 119883 (10)

on the degree of ill-posedness is used In this paper instead of(10) we require only a weaker assumption

1198891119909minus119887 le

100381710038171003817100381711986001199091003817100381710038171003817 le 1198892119909minus119887 119909 isin 119863 (119865) (11)

for some positive reals 119887 1198891 and 119889

2

Note that (11) is simpler than that of (10) Now we define119891 and 119892 by

119891 (119905) = min 1198891199051 119889119905

2 119892 (119905) = max 119889119905

1 119889119905

2

119905 isin R |119905| le 1(12)

The following proposition is used for further analysis

Proposition 2 (cf see [29 Proposition 21]) For 119904 gt 0 and|]| le 1

119891 (]) 119909minus](119904+119887) le100381710038171003817100381710038171003817(119860lowast

119904119860119904)]2119909100381710038171003817100381710038171003817le 119892 (]) 119909minus](119904+119887) 119909 isin 119867

(13)

Let us define a few parameters essential for the conver-gence analysis Let

1205952(119904) =

119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887)) 120595

2(119904) =

1

119891 (119904 (119904 + 119887))

119890120575

119899120572119904=10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 forall119899 ge 0

1205750lt1205731198872(119904+119887)

211989601205952(119904)

(1205952 (119904) +

1205722

0

21205732)

1003817100381710038171003817119909 minus 11990901003817100381710038171003817 le 120588

(14)

with

120588 ltminus1

1198960

+1

11989601205952 (119904)

times radic1205952(119904) [(

1205722

21205732+ 1205952(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

120574120588= 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

1198960

21205882+ 120588)

(15)

Journal of Mathematics 3

Further let 120574120588lt 1205722

0411989601205732 and

1199031=

120572 + radic1205722 minus 4119896

01205741205881205732

21198960120573

1199032= min

1

1205952(119904) 1198960

120572 minus radic1205722 minus 4119896

01205741205881205732

21198960120573

(16)

For 119903 isin (1199031 1199032) let

119902 = 1205952(119904) (119896

0119903 +120573 minus 120572

120573) (17)

Then 119902 lt 1

Lemma 3 Let Proposition 2 hold Then for all ℎ isin 119883 thefollowing hold

(a) (119860lowast01198600+ 120573119861119904)minus1119860lowast

01198600ℎ le 120595

2(119904)ℎ

(b) (119860lowast01198600+ 120573119861119904)minus1119861119904ℎ le 120595

2(119904)(1120573)ℎ

Proof Observe that by Proposition 2

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198601199041198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus1(119860lowast

119904119860119904) 1198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904119860119904

10038171003817100381710038171003817

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (s (119904 + 119887))100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))ℎ

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119861119904ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus110038171003817100381710038171003817

100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573

100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573ℎ

(18)

This completes the proof of the lemma

Theorem 4 Let 119890120575119899120572119904

and 119902 be as in (14) and (17) respectivelylet 119909120575119899120572119904

be as defined in (6) with 120575 isin (0 1205750] Then under

Assumption 1 and Lemma 3 the following estimates hold forall 119899 ge 0

(a) 119909120575119899+1120572119904

minus 119909120575

119899120572119904 le 119902119899120574120588

(b) 119909120575119899120572119904

isin 119861119903(1199090)

Proof If 119909120575119899120572119904

isin 119861119903(1199090) then by Assumption 1

119909120575

119899+1120572119904minus 119909120575

119899120572119904

= 119909120575

119899120572119904minus 119909120575

119899minus1120572119904minus (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ 120572119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904) ]

= (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

01198600(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

minus 119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ (120573 minus 120572) 119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)]

= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

+ (119860lowast

01198600+ 120573119861119904)minus1(120573 minus 120572) 119861

119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

= Γ1+ Γ2

(19)

where

Γ1= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

(20)

4 Journal of Mathematics

and Γ2= (119860lowast

01198600+120573119861119904)minus1(120573minus120572)119861

119904(119909120575

119899120572119904minus119909120575

119899minus1120572119904) and hence

by Assumption 1 and Lemma 3(a) we have

1003817100381710038171003817Γ11003817100381710038171003817 =10038171003817100381710038171003817minus(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600

times int

1

0

Φ(119909120575

119899minus1120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904)

1199090 119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 11988911990510038171003817100381710038171003817

le 1205952(119904) 119896011990310038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

(21)

and by Lemma 3(b)

1003817100381710038171003817Γ21003817100381710038171003817 le

120573 minus 120572

1205731205952(119904)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 (22)

Hence by (19) (21) and (22) we have

10038171003817100381710038171003817119909120575

119899+1120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le 1205952(119904) (119896

0119903 +120573 minus 120572

120573)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

= 11990210038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

le 119902119899 10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817= 119902119899119890120575

1120572119904

(23)

Next we show that 1198901205751120572119904

lt 120574120588 using Assumption 1 and

Lemma 3 Observe that

119890120575

1120572119904=10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817

=10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

0(119865 (119909120575

0120572119904) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119865 (1199090) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times [119910120575minus 119910 + 119865 (119909)

minus119865 (1199090) minus 1198600(119909 minus 119909

0) + 1198600(119909 minus 119909

0)]10038171003817100381710038171003817

le10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119910120575minus 119910)

10038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times int

1

0

(1198651015840(1199090+ 119905 (119909 minus 119909

0)) minus 119860

0)

times (119909 minus 1199090) 11988911990510038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198600(119909 minus 119909

0)10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))120573minus1198872(119904+119887)

120575 +119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

11989601205882

2

+119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))120588

le 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

11989601205882

2+ 120588)

= 120574120588lt 119903

(24)

Hence (a) follows from (23) and (24)

To prove (b) note that 1199091205751120572119904

minus 119909120575

0120572119904 le 120574120588lt 119903 Suppose

119909120575

119898120572119904isin 119861119903(1199090) for some119898 then

10038171003817100381710038171003817119909120575

119898+1120572119904minus 1199090

10038171003817100381710038171003817

le10038171003817100381710038171003817119909120575

119898+1120572minus 119909120575

119898120572119904

10038171003817100381710038171003817+ sdot sdot sdot +

10038171003817100381710038171003817119909ℎ120575

1120572119904minus 1199090

10038171003817100381710038171003817

le (119902119898+ 119902(119898minus1)

+ sdot sdot sdot + 1) 119890120575

1120572119904

le1

1 minus 119902119890120575

1120572119904

le

120574120588

1 minus 119902

lt 119903

(25)

Thus by induction 119909120575119899120572119904

isin 119861119903(1199090) for all 119899 ge 0 This proves

(b)Next we go to the main result of this section

Theorem5 Let119909120575119899120572s be as in (6) 120575 isin (0 1205750] and assumptions

ofTheorem 4 holdThen (119909120575119899120572119904) is a Cauchy sequence in 119861

119903(1199090)

and converges say to 119909120575120572119904isin 119861119903(1199090) Further119860lowast

0(119865(119909120575

120572119904)minus119910120575)+

120572119861119904(119909120575

120572119904minus 1199090) = 0 and

10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

120572119904

10038171003817100381710038171003817le 119862119902119899 (26)

where 119862 = 120574120588(1 minus 119902)

Proof Using relation (a) of Theorem 4 we obtain

10038171003817100381710038171003817119909120575

119899+119898120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

10038171003817100381710038171003817119909120575

119899+119894+1120572119904minus 119909120575

119899+119894120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

119902(119899+119894)119890120575

1120572119904

Journal of Mathematics 5

= 119902119899119890120575

1120572119904+ 119902(119899+1)

119890120575

1120572119904

+ sdot sdot sdot + 119902(119899+119898)

119890120575

1120572119904

le 119902119899(1 + 119902 + 119902

2+ sdot sdot sdot + 119902

119898) 119890120575

1120572119904

le 119902119899(1

1 minus 119902) 120574120588

le 119862119902119899

(27)

Thus 119909120575119899120572119904

is a Cauchy sequence in 119861119903(1199090) and hence it

converges say to 119909120575120572119904isin 119861119903(1199090)

Now letting 119899 rarr infin in (6) we obtain

119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 (28)

This completes the proof

The following assumption on source function and sourcecondition is required to obtain the error estimates

Assumption 6 There exists a continuous strictly monotoni-cally increasing function 120593 (0 119860

1199042] rarr (0infin) such that

the following conditions hold

(i) lim120582rarr0

120593(120582) = 0

(ii) sup120582gt0(120572120593(120582)(120582 + 120572)) le 120593(120572) for all 120582 isin (0 119860

1199042]

and

(iii) there exists 119908 isin 119883 with 119908 le 1198642 such that

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus 119909) = 120593 (119860

lowast

119904119860119904) 119908 (29)

Remark 7 If 1199090minus 119909 isin 119883

119905 for example 119909

0minus 119909119905le 1198641 for

some positive constant 1198641and 0 le 119905 le 2119904 + 119887 then we have

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus119909) = 120593(119860

lowast

119904119860119904)119908 where120593(120582) = 120582119905(119904+119887)

119908 = (119860lowast

119904119860119904)(119904minus119905)2(119904+119887)

1198611199042(119909 minus 119909

0) and 119908 le 119892((119904 minus 119905)(119904 +

119887))1198641= 1198642

Theorem 8 Let 119909120575120572119904

be the solution of (7) and supposeAssumptions 1 and 6 hold then

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952 (119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952 (119904) 1198960119903

(30)

Proof Let119872 = int1

01198651015840(119909 + 119905(119909

120575

120572119904minus 119909))119889119905 Then

119865 (119909120575

120572119904) minus 119865 (119909) = 119872(119909

120575

120572119904minus 119909) (31)

Since 119860lowast0(119865(119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 one can see that

(119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

= (119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

minus 119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) minus 120572119861

119904(119909120575

120572119904minus 1199090)

= 119860lowast

0[1198600minus119872] (119909

120575

120572119904minus 119909) + 119860

lowast

0(119910120575minus 119910)

+ 120572119861119904(1199090minus 119909)

119909120575

120572119904minus 119909 = (119860

lowast

01198600+ 120572119861119904)minus1

times [119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

+119860lowast

0(119910120575minus 119910) + 120572119861

119904(1199090minus 119909)]

= 1199041+ 1199042+ 1199043

(32)

where

1199041= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

1199042= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(119910120575minus 119910)

1199043= (119860lowast

01198600+ 120572119861119904)minus1120572119861119904(1199090minus 119909)

(33)

Note that by Assumption 1 and Lemma 3100381710038171003817100381711990411003817100381710038171003817 le 1205952 (119904) 1198960119903

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817 (34)

by Proposition 2100381710038171003817100381711990421003817100381710038171003817 le 1205952(119904)120572

minus1198872(119904+119887)120575 (35)

and by Assumption 6100381710038171003817100381711990431003817100381710038171003817 le 1205952(119904)120593 (120572)

(36)

Hence by (34)ndash(36) and (32) we have

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(37)

This completes the proof of the theorem

21 Error Bounds under Source Conditions

Theorem 9 Let 119909120575119899120572119904

be as in (6) If assumptions inTheorems5 and 8 hold then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119902119899+

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(38)

where 119862 is as in Theorem 5 Further if 119899120575= min119899 119902119899 le

120572minus1198872(119904+119887)

120575 then10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119904(120593 (120572) + 120572

minus1198872(119904+119887)120575) (39)

where 119862119904= 119862 + (120595

2(119904)(1 minus 120595

2(119904)1198960119903))

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Journal of Mathematics 3

Further let 120574120588lt 1205722

0411989601205732 and

1199031=

120572 + radic1205722 minus 4119896

01205741205881205732

21198960120573

1199032= min

1

1205952(119904) 1198960

120572 minus radic1205722 minus 4119896

01205741205881205732

21198960120573

(16)

For 119903 isin (1199031 1199032) let

119902 = 1205952(119904) (119896

0119903 +120573 minus 120572

120573) (17)

Then 119902 lt 1

Lemma 3 Let Proposition 2 hold Then for all ℎ isin 119883 thefollowing hold

(a) (119860lowast01198600+ 120573119861119904)minus1119860lowast

01198600ℎ le 120595

2(119904)ℎ

(b) (119860lowast01198600+ 120573119861119904)minus1119861119904ℎ le 120595

2(119904)(1120573)ℎ

Proof Observe that by Proposition 2

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198601199041198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus1(119860lowast

119904119860119904) 1198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904119860119904

10038171003817100381710038171003817

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (s (119904 + 119887))100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))ℎ

10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119861119904ℎ10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

(119860lowast

119904119860119904+ 120573119868)minus11198611199042ℎ100381710038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))

times10038171003817100381710038171003817(119860lowast

119904119860119904+ 120573119868)minus110038171003817100381710038171003817

100381710038171003817100381710038171003817(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042ℎ100381710038171003817100381710038171003817

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573

100381710038171003817100381710038171198611199042ℎ10038171003817100381710038171003817minus119904

le119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

1

120573ℎ

(18)

This completes the proof of the lemma

Theorem 4 Let 119890120575119899120572119904

and 119902 be as in (14) and (17) respectivelylet 119909120575119899120572119904

be as defined in (6) with 120575 isin (0 1205750] Then under

Assumption 1 and Lemma 3 the following estimates hold forall 119899 ge 0

(a) 119909120575119899+1120572119904

minus 119909120575

119899120572119904 le 119902119899120574120588

(b) 119909120575119899120572119904

isin 119861119903(1199090)

Proof If 119909120575119899120572119904

isin 119861119903(1199090) then by Assumption 1

119909120575

119899+1120572119904minus 119909120575

119899120572119904

= 119909120575

119899120572119904minus 119909120575

119899minus1120572119904minus (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ 120572119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904) ]

= (119860lowast

01198600+ 120573119861119904)minus1

times [119860lowast

01198600(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

minus 119860lowast

0(119865 (119909120575

119899120572119904) minus 119865 (119909

120575

119899minus1120572119904))

+ (120573 minus 120572) 119861119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)]

= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

+ (119860lowast

01198600+ 120573119861119904)minus1(120573 minus 120572) 119861

119904(119909120575

119899120572119904minus 119909120575

119899minus1120572119904)

= Γ1+ Γ2

(19)

where

Γ1= (119860lowast

01198600+ 120573119861119904)minus1119860lowast

0

times int

1

0

[1198600minus 1198651015840(119909120575

119899120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904))]

times (119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 119889119905

(20)

4 Journal of Mathematics

and Γ2= (119860lowast

01198600+120573119861119904)minus1(120573minus120572)119861

119904(119909120575

119899120572119904minus119909120575

119899minus1120572119904) and hence

by Assumption 1 and Lemma 3(a) we have

1003817100381710038171003817Γ11003817100381710038171003817 =10038171003817100381710038171003817minus(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600

times int

1

0

Φ(119909120575

119899minus1120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904)

1199090 119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 11988911990510038171003817100381710038171003817

le 1205952(119904) 119896011990310038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

(21)

and by Lemma 3(b)

1003817100381710038171003817Γ21003817100381710038171003817 le

120573 minus 120572

1205731205952(119904)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 (22)

Hence by (19) (21) and (22) we have

10038171003817100381710038171003817119909120575

119899+1120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le 1205952(119904) (119896

0119903 +120573 minus 120572

120573)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

= 11990210038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

le 119902119899 10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817= 119902119899119890120575

1120572119904

(23)

Next we show that 1198901205751120572119904

lt 120574120588 using Assumption 1 and

Lemma 3 Observe that

119890120575

1120572119904=10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817

=10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

0(119865 (119909120575

0120572119904) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119865 (1199090) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times [119910120575minus 119910 + 119865 (119909)

minus119865 (1199090) minus 1198600(119909 minus 119909

0) + 1198600(119909 minus 119909

0)]10038171003817100381710038171003817

le10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119910120575minus 119910)

10038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times int

1

0

(1198651015840(1199090+ 119905 (119909 minus 119909

0)) minus 119860

0)

times (119909 minus 1199090) 11988911990510038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198600(119909 minus 119909

0)10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))120573minus1198872(119904+119887)

120575 +119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

11989601205882

2

+119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))120588

le 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

11989601205882

2+ 120588)

= 120574120588lt 119903

(24)

Hence (a) follows from (23) and (24)

To prove (b) note that 1199091205751120572119904

minus 119909120575

0120572119904 le 120574120588lt 119903 Suppose

119909120575

119898120572119904isin 119861119903(1199090) for some119898 then

10038171003817100381710038171003817119909120575

119898+1120572119904minus 1199090

10038171003817100381710038171003817

le10038171003817100381710038171003817119909120575

119898+1120572minus 119909120575

119898120572119904

10038171003817100381710038171003817+ sdot sdot sdot +

10038171003817100381710038171003817119909ℎ120575

1120572119904minus 1199090

10038171003817100381710038171003817

le (119902119898+ 119902(119898minus1)

+ sdot sdot sdot + 1) 119890120575

1120572119904

le1

1 minus 119902119890120575

1120572119904

le

120574120588

1 minus 119902

lt 119903

(25)

Thus by induction 119909120575119899120572119904

isin 119861119903(1199090) for all 119899 ge 0 This proves

(b)Next we go to the main result of this section

Theorem5 Let119909120575119899120572s be as in (6) 120575 isin (0 1205750] and assumptions

ofTheorem 4 holdThen (119909120575119899120572119904) is a Cauchy sequence in 119861

119903(1199090)

and converges say to 119909120575120572119904isin 119861119903(1199090) Further119860lowast

0(119865(119909120575

120572119904)minus119910120575)+

120572119861119904(119909120575

120572119904minus 1199090) = 0 and

10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

120572119904

10038171003817100381710038171003817le 119862119902119899 (26)

where 119862 = 120574120588(1 minus 119902)

Proof Using relation (a) of Theorem 4 we obtain

10038171003817100381710038171003817119909120575

119899+119898120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

10038171003817100381710038171003817119909120575

119899+119894+1120572119904minus 119909120575

119899+119894120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

119902(119899+119894)119890120575

1120572119904

Journal of Mathematics 5

= 119902119899119890120575

1120572119904+ 119902(119899+1)

119890120575

1120572119904

+ sdot sdot sdot + 119902(119899+119898)

119890120575

1120572119904

le 119902119899(1 + 119902 + 119902

2+ sdot sdot sdot + 119902

119898) 119890120575

1120572119904

le 119902119899(1

1 minus 119902) 120574120588

le 119862119902119899

(27)

Thus 119909120575119899120572119904

is a Cauchy sequence in 119861119903(1199090) and hence it

converges say to 119909120575120572119904isin 119861119903(1199090)

Now letting 119899 rarr infin in (6) we obtain

119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 (28)

This completes the proof

The following assumption on source function and sourcecondition is required to obtain the error estimates

Assumption 6 There exists a continuous strictly monotoni-cally increasing function 120593 (0 119860

1199042] rarr (0infin) such that

the following conditions hold

(i) lim120582rarr0

120593(120582) = 0

(ii) sup120582gt0(120572120593(120582)(120582 + 120572)) le 120593(120572) for all 120582 isin (0 119860

1199042]

and

(iii) there exists 119908 isin 119883 with 119908 le 1198642 such that

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus 119909) = 120593 (119860

lowast

119904119860119904) 119908 (29)

Remark 7 If 1199090minus 119909 isin 119883

119905 for example 119909

0minus 119909119905le 1198641 for

some positive constant 1198641and 0 le 119905 le 2119904 + 119887 then we have

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus119909) = 120593(119860

lowast

119904119860119904)119908 where120593(120582) = 120582119905(119904+119887)

119908 = (119860lowast

119904119860119904)(119904minus119905)2(119904+119887)

1198611199042(119909 minus 119909

0) and 119908 le 119892((119904 minus 119905)(119904 +

119887))1198641= 1198642

Theorem 8 Let 119909120575120572119904

be the solution of (7) and supposeAssumptions 1 and 6 hold then

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952 (119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952 (119904) 1198960119903

(30)

Proof Let119872 = int1

01198651015840(119909 + 119905(119909

120575

120572119904minus 119909))119889119905 Then

119865 (119909120575

120572119904) minus 119865 (119909) = 119872(119909

120575

120572119904minus 119909) (31)

Since 119860lowast0(119865(119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 one can see that

(119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

= (119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

minus 119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) minus 120572119861

119904(119909120575

120572119904minus 1199090)

= 119860lowast

0[1198600minus119872] (119909

120575

120572119904minus 119909) + 119860

lowast

0(119910120575minus 119910)

+ 120572119861119904(1199090minus 119909)

119909120575

120572119904minus 119909 = (119860

lowast

01198600+ 120572119861119904)minus1

times [119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

+119860lowast

0(119910120575minus 119910) + 120572119861

119904(1199090minus 119909)]

= 1199041+ 1199042+ 1199043

(32)

where

1199041= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

1199042= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(119910120575minus 119910)

1199043= (119860lowast

01198600+ 120572119861119904)minus1120572119861119904(1199090minus 119909)

(33)

Note that by Assumption 1 and Lemma 3100381710038171003817100381711990411003817100381710038171003817 le 1205952 (119904) 1198960119903

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817 (34)

by Proposition 2100381710038171003817100381711990421003817100381710038171003817 le 1205952(119904)120572

minus1198872(119904+119887)120575 (35)

and by Assumption 6100381710038171003817100381711990431003817100381710038171003817 le 1205952(119904)120593 (120572)

(36)

Hence by (34)ndash(36) and (32) we have

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(37)

This completes the proof of the theorem

21 Error Bounds under Source Conditions

Theorem 9 Let 119909120575119899120572119904

be as in (6) If assumptions inTheorems5 and 8 hold then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119902119899+

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(38)

where 119862 is as in Theorem 5 Further if 119899120575= min119899 119902119899 le

120572minus1198872(119904+119887)

120575 then10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119904(120593 (120572) + 120572

minus1198872(119904+119887)120575) (39)

where 119862119904= 119862 + (120595

2(119904)(1 minus 120595

2(119904)1198960119903))

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

4 Journal of Mathematics

and Γ2= (119860lowast

01198600+120573119861119904)minus1(120573minus120572)119861

119904(119909120575

119899120572119904minus119909120575

119899minus1120572119904) and hence

by Assumption 1 and Lemma 3(a) we have

1003817100381710038171003817Γ11003817100381710038171003817 =10038171003817100381710038171003817minus(119860lowast

01198600+ 120573119861119904)minus1119860lowast

01198600

times int

1

0

Φ(119909120575

119899minus1120572119904+ 119905 (119909

120575

119899120572119904minus 119909120575

119899minus1120572119904)

1199090 119909120575

119899120572119904minus 119909120575

119899minus1120572119904) 11988911990510038171003817100381710038171003817

le 1205952(119904) 119896011990310038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

(21)

and by Lemma 3(b)

1003817100381710038171003817Γ21003817100381710038171003817 le

120573 minus 120572

1205731205952(119904)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817 (22)

Hence by (19) (21) and (22) we have

10038171003817100381710038171003817119909120575

119899+1120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le 1205952(119904) (119896

0119903 +120573 minus 120572

120573)10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

= 11990210038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

119899minus1120572119904

10038171003817100381710038171003817

le 119902119899 10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817= 119902119899119890120575

1120572119904

(23)

Next we show that 1198901205751120572119904

lt 120574120588 using Assumption 1 and

Lemma 3 Observe that

119890120575

1120572119904=10038171003817100381710038171003817119909120575

1120572119904minus 119909120575

0120572119904

10038171003817100381710038171003817

=10038171003817100381710038171003817(119860lowast

01198600+ 120573119861119904)minus1119860lowast

0(119865 (119909120575

0120572119904) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119865 (1199090) minus 119910120575)10038171003817100381710038171003817

=10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times [119910120575minus 119910 + 119865 (119909)

minus119865 (1199090) minus 1198600(119909 minus 119909

0) + 1198600(119909 minus 119909

0)]10038171003817100381710038171003817

le10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904(119910120575minus 119910)

10038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

119904

times int

1

0

(1198651015840(1199090+ 119905 (119909 minus 119909

0)) minus 119860

0)

times (119909 minus 1199090) 11988911990510038171003817100381710038171003817

+10038171003817100381710038171003817119861minus1199042(119860lowast

119904119860119904+ 120573119868)minus1119860lowast

1199041198600(119909 minus 119909

0)10038171003817100381710038171003817

le1

119891 (119904 (119904 + 119887))120573minus1198872(119904+119887)

120575 +119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))

11989601205882

2

+119892 (119904 (119904 + 119887))

119891 (119904 (119904 + 119887))120588

le 1205952(119904)120573minus1198872(119904+119887)

120575 + 1205952(119904) (

11989601205882

2+ 120588)

= 120574120588lt 119903

(24)

Hence (a) follows from (23) and (24)

To prove (b) note that 1199091205751120572119904

minus 119909120575

0120572119904 le 120574120588lt 119903 Suppose

119909120575

119898120572119904isin 119861119903(1199090) for some119898 then

10038171003817100381710038171003817119909120575

119898+1120572119904minus 1199090

10038171003817100381710038171003817

le10038171003817100381710038171003817119909120575

119898+1120572minus 119909120575

119898120572119904

10038171003817100381710038171003817+ sdot sdot sdot +

10038171003817100381710038171003817119909ℎ120575

1120572119904minus 1199090

10038171003817100381710038171003817

le (119902119898+ 119902(119898minus1)

+ sdot sdot sdot + 1) 119890120575

1120572119904

le1

1 minus 119902119890120575

1120572119904

le

120574120588

1 minus 119902

lt 119903

(25)

Thus by induction 119909120575119899120572119904

isin 119861119903(1199090) for all 119899 ge 0 This proves

(b)Next we go to the main result of this section

Theorem5 Let119909120575119899120572s be as in (6) 120575 isin (0 1205750] and assumptions

ofTheorem 4 holdThen (119909120575119899120572119904) is a Cauchy sequence in 119861

119903(1199090)

and converges say to 119909120575120572119904isin 119861119903(1199090) Further119860lowast

0(119865(119909120575

120572119904)minus119910120575)+

120572119861119904(119909120575

120572119904minus 1199090) = 0 and

10038171003817100381710038171003817119909120575

119899120572119904minus 119909120575

120572119904

10038171003817100381710038171003817le 119862119902119899 (26)

where 119862 = 120574120588(1 minus 119902)

Proof Using relation (a) of Theorem 4 we obtain

10038171003817100381710038171003817119909120575

119899+119898120572119904minus 119909120575

119899120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

10038171003817100381710038171003817119909120575

119899+119894+1120572119904minus 119909120575

119899+119894120572119904

10038171003817100381710038171003817

le

119894=119898minus1

sum

119894=0

119902(119899+119894)119890120575

1120572119904

Journal of Mathematics 5

= 119902119899119890120575

1120572119904+ 119902(119899+1)

119890120575

1120572119904

+ sdot sdot sdot + 119902(119899+119898)

119890120575

1120572119904

le 119902119899(1 + 119902 + 119902

2+ sdot sdot sdot + 119902

119898) 119890120575

1120572119904

le 119902119899(1

1 minus 119902) 120574120588

le 119862119902119899

(27)

Thus 119909120575119899120572119904

is a Cauchy sequence in 119861119903(1199090) and hence it

converges say to 119909120575120572119904isin 119861119903(1199090)

Now letting 119899 rarr infin in (6) we obtain

119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 (28)

This completes the proof

The following assumption on source function and sourcecondition is required to obtain the error estimates

Assumption 6 There exists a continuous strictly monotoni-cally increasing function 120593 (0 119860

1199042] rarr (0infin) such that

the following conditions hold

(i) lim120582rarr0

120593(120582) = 0

(ii) sup120582gt0(120572120593(120582)(120582 + 120572)) le 120593(120572) for all 120582 isin (0 119860

1199042]

and

(iii) there exists 119908 isin 119883 with 119908 le 1198642 such that

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus 119909) = 120593 (119860

lowast

119904119860119904) 119908 (29)

Remark 7 If 1199090minus 119909 isin 119883

119905 for example 119909

0minus 119909119905le 1198641 for

some positive constant 1198641and 0 le 119905 le 2119904 + 119887 then we have

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus119909) = 120593(119860

lowast

119904119860119904)119908 where120593(120582) = 120582119905(119904+119887)

119908 = (119860lowast

119904119860119904)(119904minus119905)2(119904+119887)

1198611199042(119909 minus 119909

0) and 119908 le 119892((119904 minus 119905)(119904 +

119887))1198641= 1198642

Theorem 8 Let 119909120575120572119904

be the solution of (7) and supposeAssumptions 1 and 6 hold then

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952 (119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952 (119904) 1198960119903

(30)

Proof Let119872 = int1

01198651015840(119909 + 119905(119909

120575

120572119904minus 119909))119889119905 Then

119865 (119909120575

120572119904) minus 119865 (119909) = 119872(119909

120575

120572119904minus 119909) (31)

Since 119860lowast0(119865(119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 one can see that

(119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

= (119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

minus 119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) minus 120572119861

119904(119909120575

120572119904minus 1199090)

= 119860lowast

0[1198600minus119872] (119909

120575

120572119904minus 119909) + 119860

lowast

0(119910120575minus 119910)

+ 120572119861119904(1199090minus 119909)

119909120575

120572119904minus 119909 = (119860

lowast

01198600+ 120572119861119904)minus1

times [119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

+119860lowast

0(119910120575minus 119910) + 120572119861

119904(1199090minus 119909)]

= 1199041+ 1199042+ 1199043

(32)

where

1199041= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

1199042= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(119910120575minus 119910)

1199043= (119860lowast

01198600+ 120572119861119904)minus1120572119861119904(1199090minus 119909)

(33)

Note that by Assumption 1 and Lemma 3100381710038171003817100381711990411003817100381710038171003817 le 1205952 (119904) 1198960119903

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817 (34)

by Proposition 2100381710038171003817100381711990421003817100381710038171003817 le 1205952(119904)120572

minus1198872(119904+119887)120575 (35)

and by Assumption 6100381710038171003817100381711990431003817100381710038171003817 le 1205952(119904)120593 (120572)

(36)

Hence by (34)ndash(36) and (32) we have

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(37)

This completes the proof of the theorem

21 Error Bounds under Source Conditions

Theorem 9 Let 119909120575119899120572119904

be as in (6) If assumptions inTheorems5 and 8 hold then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119902119899+

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(38)

where 119862 is as in Theorem 5 Further if 119899120575= min119899 119902119899 le

120572minus1198872(119904+119887)

120575 then10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119904(120593 (120572) + 120572

minus1198872(119904+119887)120575) (39)

where 119862119904= 119862 + (120595

2(119904)(1 minus 120595

2(119904)1198960119903))

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Journal of Mathematics 5

= 119902119899119890120575

1120572119904+ 119902(119899+1)

119890120575

1120572119904

+ sdot sdot sdot + 119902(119899+119898)

119890120575

1120572119904

le 119902119899(1 + 119902 + 119902

2+ sdot sdot sdot + 119902

119898) 119890120575

1120572119904

le 119902119899(1

1 minus 119902) 120574120588

le 119862119902119899

(27)

Thus 119909120575119899120572119904

is a Cauchy sequence in 119861119903(1199090) and hence it

converges say to 119909120575120572119904isin 119861119903(1199090)

Now letting 119899 rarr infin in (6) we obtain

119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 (28)

This completes the proof

The following assumption on source function and sourcecondition is required to obtain the error estimates

Assumption 6 There exists a continuous strictly monotoni-cally increasing function 120593 (0 119860

1199042] rarr (0infin) such that

the following conditions hold

(i) lim120582rarr0

120593(120582) = 0

(ii) sup120582gt0(120572120593(120582)(120582 + 120572)) le 120593(120572) for all 120582 isin (0 119860

1199042]

and

(iii) there exists 119908 isin 119883 with 119908 le 1198642 such that

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus 119909) = 120593 (119860

lowast

119904119860119904) 119908 (29)

Remark 7 If 1199090minus 119909 isin 119883

119905 for example 119909

0minus 119909119905le 1198641 for

some positive constant 1198641and 0 le 119905 le 2119904 + 119887 then we have

(119860lowast

119904119860119904)1199042(119904+119887)

1198611199042(1199090minus119909) = 120593(119860

lowast

119904119860119904)119908 where120593(120582) = 120582119905(119904+119887)

119908 = (119860lowast

119904119860119904)(119904minus119905)2(119904+119887)

1198611199042(119909 minus 119909

0) and 119908 le 119892((119904 minus 119905)(119904 +

119887))1198641= 1198642

Theorem 8 Let 119909120575120572119904

be the solution of (7) and supposeAssumptions 1 and 6 hold then

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952 (119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952 (119904) 1198960119903

(30)

Proof Let119872 = int1

01198651015840(119909 + 119905(119909

120575

120572119904minus 119909))119889119905 Then

119865 (119909120575

120572119904) minus 119865 (119909) = 119872(119909

120575

120572119904minus 119909) (31)

Since 119860lowast0(119865(119909120575

120572119904) minus 119910120575) + 120572119861

119904(119909120575

120572119904minus 1199090) = 0 one can see that

(119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

= (119860lowast

01198600+ 120572119861119904) (119909120575

120572119904minus 119909)

minus 119860lowast

0(119865 (119909120575

120572119904) minus 119910120575) minus 120572119861

119904(119909120575

120572119904minus 1199090)

= 119860lowast

0[1198600minus119872] (119909

120575

120572119904minus 119909) + 119860

lowast

0(119910120575minus 119910)

+ 120572119861119904(1199090minus 119909)

119909120575

120572119904minus 119909 = (119860

lowast

01198600+ 120572119861119904)minus1

times [119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

+119860lowast

0(119910120575minus 119910) + 120572119861

119904(1199090minus 119909)]

= 1199041+ 1199042+ 1199043

(32)

where

1199041= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(1198600minus119872) (119909

120575

120572119904minus 119909)

1199042= (119860lowast

01198600+ 120572119861119904)minus1119860lowast

0(119910120575minus 119910)

1199043= (119860lowast

01198600+ 120572119861119904)minus1120572119861119904(1199090minus 119909)

(33)

Note that by Assumption 1 and Lemma 3100381710038171003817100381711990411003817100381710038171003817 le 1205952 (119904) 1198960119903

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817 (34)

by Proposition 2100381710038171003817100381711990421003817100381710038171003817 le 1205952(119904)120572

minus1198872(119904+119887)120575 (35)

and by Assumption 6100381710038171003817100381711990431003817100381710038171003817 le 1205952(119904)120593 (120572)

(36)

Hence by (34)ndash(36) and (32) we have

10038171003817100381710038171003817119909120575

120572119904minus 11990910038171003817100381710038171003817le

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(37)

This completes the proof of the theorem

21 Error Bounds under Source Conditions

Theorem 9 Let 119909120575119899120572119904

be as in (6) If assumptions inTheorems5 and 8 hold then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119902119899+

1205952(119904) (120593 (120572) + 120572

minus1198872(119904+119887)120575)

1 minus 1205952(119904) 1198960119903

(38)

where 119862 is as in Theorem 5 Further if 119899120575= min119899 119902119899 le

120572minus1198872(119904+119887)

120575 then10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119862119904(120593 (120572) + 120572

minus1198872(119904+119887)120575) (39)

where 119862119904= 119862 + (120595

2(119904)(1 minus 120595

2(119904)1198960119903))

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

6 Journal of Mathematics

0 01 02 03 04 05 06 07 08 090

01

02

03

04

05

06

07

08

Exact solutionApproximate solution

Exact solutionApproximate solution

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

0 02 04 06 08 10

02

04

06

08

1

12

14n = 8 n = 16

n = 32 n = 64

Figure 1 Curves of the exact and approximate solutions for 119899 = 8 16 32 64

22 A Priori Choice of the Parameter The error estimate120593(120572) + 120572

minus1198872(119904+119887)120575 in Theorem 9 attains minimum for the

choice 120572 = 120572(120575 119904 119887) which satisfies 120593(120572) = 120572minus1198872(119904+119887)

120575Clearly 120572(120575 119904 119887) = 120593minus1(120595minus1s119887 (120575)) where

120595119904119887 (120582) = 120582[120593

minus1(120582)]1198872(119904+119887)

0 lt 120582 le1003817100381710038171003817119860 1199041003817100381710038171003817

2 (40)

Thus we have the following theorem

Theorem 10 Suppose that all assumptions of Theorems 5 and8 are fulfilled For 120575 gt 0 let 120572(120575 119904 119887) = 120593minus1(120595minus1

119904119887(120575)) and let

119899120575be as in Theorem 9 Then

10038171003817100381710038171003817119909 minus 119909120575

119899120572119904

10038171003817100381710038171003817le 119874 (120595

minus1

119904119887(120575)) (41)

23 Adaptive Scheme and Stopping Rule In this subsectionwe consider the adaptive scheme suggested by Pereverzev andSchock in [17] modified suitably for choosing the parameter

120572 which does not involve even the regularization method inan explicit manner

Let 119894 isin 0 1 2 119873 and 120572119894= 1205831198941205720 where 120583 = 1205782(1+119904119887)

120578 gt 1 and 1205720= 1205752(1+119904119887) Let 119899

119894= min119899 119902119899 le 120572minus1198872(119904+119887)

119894120575

and let 119909120575119899119894 120572119894 119904

be as defined in (6) with 120572 = 120572119894and 119899 = 119899

119894

Then fromTheorem 9 we have10038171003817100381710038171003817119909 minus 119909120575

119899119894 120572119894 119904

10038171003817100381710038171003817le 119862119904(120593 (120572119894) + 120572minus1198872(119904+119887)

119894120575) (42)

Further let

119897 = max 119894 120593 (120572119894) le 120572minus1198872(119904+119887)

119894120575 lt 119873 (43)

119896 = max 119894 100381710038171003817100381710038171003817119909120575

119899119894 120572119894119904minus 119909120575

119899119895 120572119895119904

100381710038171003817100381710038171003817le 4119862119904120572minus1198872(119904+119887)

119895120575

119895 = 0 1 2 119894 minus 1

(44)

where 119862119904is as in Theorem 9 The proof of the following

theorem is analogous to the proof of Theorem 44 in [31] sowe omit the details

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Journal of Mathematics 7

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

Exact solutionApproximate solution

n = 128 n = 256

n = 512 n = 1024

Figure 2 Curves of the exact and approximate solutions for 119899 = 128 256 512 1024

Table 1 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 1

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10606 09249 2924416 2 9 02857 10033 3172432 2 9 01431 10417 3294064 2 9 01429 10608 33543128 2 9 01428 10704 33847256 2 9 01428 10754 34005512 2 9 01428 10784 340981024 2 9 01428 10807 34172

Theorem 11 Let 119909120575119899120572119904

be as in (6)with 120572 = 120572119894and 120575 isin (0 120575

0]

and assumptions in Theorem 9 hold Let 119897 and 119896 be as definedin (43) and (44) respectively Then 119897 le 119896 and

10038171003817100381710038171003817119909 minus 119909120575

119899119896120572119896119904

10038171003817100381710038171003817le 6119862119904120578 (120595minus1

119904119887(120575)) (45)

3 Implementation of the Method

Finally the balancing algorithm associated with the choice ofthe parameter specified in Theorem 11 involves the followingsteps

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

8 Journal of Mathematics

Table 2 Iterations and corresponding error estimates with 120575 = 18119864 minus 5 120590 = 01

119899 119896 119899119896

120572 119909120575

119899119896 120572119904minus 119909

119909120575

119899119896 120572119904minus 119909

(120575)15

8 7 21 10605 09249 2924616 2 9 02856 10033 3172632 2 9 01431 10417 3294264 2 9 01429 10608 33546128 2 9 01428 10704 33850256 2 9 01428 10754 34008512 2 9 01428 10784 341011024 2 9 01428 10807 34175

(i) choose 1205720gt 0 such that 120575

0lt (1205731198872(119904+119887)

211989601205952(119904))

(1205952(119904) + (120572

2

021205732)) and 120578 gt 1

(ii) choose119873 big enough but not too large and 120572119894= 1205831198941205720

119894 = 0 1 2 119873 where 120583 = 1205782(1+119904119886)(iii) choose 120588 lt (minus1119896

0) + (1119896

01205952(119904))

radic1205952(119904)[((120572221205732) + 120595

2(119904)) minus 2119896

01205952(119904)120573minus1198872(119904+119887)120575]

Algorithm 1

(1) set 119894 = 0(2) choose 119899

119894= min119899 119902119899 le 120572minus1198862(119904+119886)

119894120575

(3) solve 119909120575119899119894 120572119894119904

by using the iteration (6)

(4) if 119909120575119899119894 120572119894119904

minus 119909120575

119899119895120572119895119904 gt 4119862

119904120572minus1198872(119904+119887)

119895120575 119895 lt 119894 then take

119896 = 119894 minus 1(5) else set 119894 = 119894 + 1 and return to Step (2)

4 Numerical Example

Example 1 In this example we consider a nonlinear integraloperator 119865 119863(119865) sub 1198712(0 1) rarr 119871

2(0 1) defined by

119865 (119909) (119905) = int

1

0

119896 (119905 119904) 119909(119904)3119889119904 = 119891 (119905) (46)

with

119896 (119905 119904) = (1 minus 119905) 119904 0 le 119904 le 119905 le 1

(1 minus 119904) 119905 0 le 119905 le 119904 le 1(47)

The Frechet derivative of 119865 is given by

1198651015840(119906) 119908 = 3int

1

0

119896 (119905 119904) (119906 (119904))2119908 (119904) 119889119904 (48)

In our computation we take 119910(119905) = (119905minus11990511)110 and 119910120575 =119910 + 120590(119910119890)119890 where 119890 = (119890

119894) is a random vector with 119890

119894sim

alefsym(0 1) and 120590 gt 0 is a constant [26] Then the exact solution

119909 (119905) = 1199053 (49)

We take 119871 119863 sub 1198712(0 1) rarr 1198712(0 1) as

119871119909 =

infin

sum

119896=1

119896 ⟨119909 119890119896⟩ 119890119896

with 119890119896(119905) = radic2 sin (119896120587119905) (50)

1199090 (119905) = 119905

3+119905

15(51)

as our initial guess so that the function 1199090minus 119909 satisfies

the source condition 1199090minus 119909119905le 119864 119905 isin [0 12) (see [20

Proposition 53]) Thus we expect to have an accuracy oforder at least 119874(12057515)

As in [26] we use the (119899 119899)matrix

119861 = 11986112

2with 119861

2=(119899 + 1)

2

1205872(

2 minus1

minus1 d dd d minus1

minus1 2

) (52)

as a discrete approximation of the first-order differentialoperator (50)

We choose 1205720= 00171 120573 = 119 119896

0= 1 119904 = 2 and

119902 = 09The results of the computation are presented inTables1 and 2 The plots of the exact and the approximate solutionobtained with 120575 = 18119864 minus 5 are given in Figures 1 and 2

The last column of Tables 1 and 2 shows that the error119909120575

120572119896 119904minus 119909 is of 119874(12057515)

5 Conclusion

In this paper we present an iterative regularization methodfor obtaining an approximate solution of a nonlinear ill-posedoperator equation 119865(119909) = 119910 in the Hilbert scale setting Here119865 119863(119865) sub 119883 rarr 119884 is a nonlinear operator and we assumethat the available data is 119910120575 in place of exact data 119910 Theconvergence analysis was based on the center-type Lipschitzcondition We considered a Hilbert scale (119883

119905)119905isinR generated

by 119861 where 119861 119863(119861) sub 119883 rarr 119883 is a linear unboundedself-adjoint densely defined and strictly positive operator on119883 For choosing the regularization parameter 120572 the adaptivescheme considered by Pereverzev and Schock in [17] wasused Finally a numerical example is presented in support ofour method which is found to be efficient

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Journal of Mathematics 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Ms Monnanda Erappa Shobha thanks NBHM DAE Gov-ernment of India for the financial support

References

[1] I K Argyros and S Hilout ldquoA convergence analysis fordirectional two-step Newton methodsrdquo Numerical Algorithmsvol 55 no 4 pp 503ndash528 2010

[2] I K Argyros and S Hilout ldquoWeaker conditions for theconvergence of Newtonrsquos methodrdquo Journal of Complexity vol28 no 3 pp 364ndash387 2012

[3] I K Argyros Y J Cho and S Hilout Numerical Methods forEquations and its Applications CRC Press Taylor and FrancisNew York NY USA 2012

[4] A B Bakushinsky and M Y Kokurin Iterative Methods forApproximate Solution of Inverse Problems Springer DordrechtThe Netherlands 2004

[5] H W Engl K Kunisch and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Dordrecht TheNetherlands 1996

[6] H W Engl ldquoRegularization methods for the stable solution ofinverse problemsrdquo Surveys on Mathematics for Industry vol 3no 2 pp 71ndash143 1993

[7] H W Engl K Kunisch and A Neubauer ldquoConvergence ratesfor Tikhonov regularisation of nonlinear ill-posed problemsrdquoInverse Problems vol 5 no 4 pp 523ndash540 1989

[8] S George ldquoNewton-type iteration for Tikhonov regularizationof nonlinear ill-posed problemsrdquo Journal of Mathematics vol2013 Article ID 439316 9 pages 2013

[9] M Hanke ldquoA regularizing Levenberg-Marquardt scheme withapplications to inverse groundwater filtration problemsrdquo InverseProblems vol 13 no 1 pp 79ndash95 1997

[10] B Kaltenbacher ldquoA note on logarithmic convergence rates fornonlinear Tikhonov regularizationrdquo Journal of Inverse and Ill-Posed Problems vol 16 no 1 pp 79ndash88 2008

[11] B Kaltenbacher A Neubauer and O Scherzer Iterative Regu-larizationMethods for Nonlinear Ill-Posed Porblems de GruyterBerlin Germany 2008

[12] C T Kelley Iterative Methods for Linear and Nonlinear Equa-tions SIAM Philadelphia Pa USA 1995

[13] Q Jin ldquoOn a regularized Levenberg-Marquardt method forsolving nonlinear inverse problemsrdquo Numerische Mathematikvol 115 no 2 pp 229ndash259 2010

[14] U Tautenhahn ldquoOn themethod of Lavrentiev regularization fornonlinear ill-posed problemsrdquo Inverse Problems vol 18 no 1pp 191ndash207 2002

[15] V Vasin ldquoIrregular nonlinear operator equations Tikhonovrsquosregularization and iterative approximationrdquo Journal of Inverseand Ill-Posed Problems vol 21 no 1 pp 109ndash123 2013

[16] V Vasin and S George ldquoExpanding the applicability ofTikhonovrsquos regularization and iterative approximation for ill-posed problemsrdquo Journal of Inverse and Ill-Posed Problems 2013

[17] S Pereverzev and E Schock ldquoOn the adaptive selection ofthe parameter in regularization of ill-posed problemsrdquo SIAMJournal on Numerical Analysis vol 43 no 5 pp 2060ndash20762005

[18] H Egger and A Neubauer ldquoPreconditioning Landweber itera-tion in Hilbert scalesrdquo Numerische Mathematik vol 101 no 4pp 643ndash662 2005

[19] Q Jin ldquoError estimates of some Newton-type methods forsolving nonlinear inverse problems in Hilbert scalesrdquo InverseProblems vol 16 no 1 pp 187ndash197 2000

[20] S Lu S V Pereverzev Y Shao and U Tautenhahn ldquoOn thegeneralized discrepancy principle for Tikhonov regularizationinHilbert scalesrdquo Journal of Integral Equations and Applicationsvol 22 no 3 pp 483ndash517 2010

[21] P Mahale and M T Nair ldquoA simplified generalized Gauss-Newton method for nonlinear ill-posed problemsrdquoMathemat-ics of Computation vol 78 no 265 pp 171ndash184 2009

[22] P Mathe and U Tautenhahn ldquoError bounds for regularizationmethods in Hilbert scales by using operator monotonicityrdquo FarEast Journal of Mathematical Sciences vol 24 no 1 pp 1ndash212007

[23] F Natterer ldquoError bounds for Tikhonov regularization inHilbert scalesrdquo Applicable Analysis vol 18 no 1-2 pp 29ndash371984

[24] A Neubauer ldquoOn Landweber iteration for nonlinear ill-posedproblems inHilbert scalesrdquoNumerischeMathematik vol 85 no2 pp 309ndash328 2000

[25] Q Jin and U Tautenhahn ldquoInexact Newton regularizationmethods inHilbert scalesrdquoNumerischeMathematik vol 117 no3 pp 555ndash579 2011

[26] Q Jin andU Tautenhahn ldquoImplicit iterationmethods inHilbertscales under general smoothness conditionsrdquo Inverse Problemsvol 27 no 4 Article ID 045012 2011

[27] S George and M T Nair ldquoError bounds and parameter choicestrategies for simplified regularization inHilbert scalesrdquo IntegralEquations and OperatorTheory vol 29 no 2 pp 231ndash242 1997

[28] U Tautenhahn ldquoOn a general regularization scheme for non-linear ill-posed problems II Regularization in Hilbert scalesrdquoInverse Problems vol 14 no 6 pp 1607ndash1616 1998

[29] U Tautenhahn ldquoError estimates for regularization methods inHilbert scalesrdquo SIAM Journal on Numerical Analysis vol 33 no6 pp 2120ndash2130 1996

[30] Q Jin ldquoOn a class of frozen regularizedGauss-Newtonmethodsfor nonlinear inverse problemsrdquo Mathematics of Computationvol 79 no 272 pp 2191ndash2211 2010

[31] S George ldquoOn convergence of regularized modified Newtonrsquosmethod for nonlinear ill-posed problemsrdquo Journal of Inverseand Ill-Posed Problems vol 18 no 2 pp 133ndash146 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Newton Type Iteration for Tikhonov ...downloads.hindawi.com/journals/jmath/2014/965097.pdf · License, which permits unrestricted use, distribution, and reproduction

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of