# Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems

Authors: Santhosh George
Journal: Journal of Mathematics
 (2013)

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## Abstract

Recently in  the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equationF(x)=y, where F:D(F)⊆X→Y is a nonlinear operator between the Hilbert spaces X and Y. The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.


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## Body

## 1. Introduction

Inverse problems have been one of the fastest growing research area in applied mathematics in the last decades. It is well known that these problems typically lead to mathematical models that are ill-posed (according to Hadamard’s definition [1]) in the sense that it is not possible to provide a unique solution.In this paper, we consider the task of approximately solving the nonlinear ill-posed equation(1)F(x)=y.
This equation and the task of solving it make sense only when placed in an appropriate framework. Throughout this paper, we will assume that F:D(F)⊆X→Y is a nonlinear operator between Hilbert spaces X and Y with inner product and corresponding norm denoted by 〈·,·〉 and ∥·∥, respectively, and y∈Y. We assume that (1) has a unique solution x^. For δ>0, let yδ∈Y be an available data with
(2)∥y-yδ∥≤δ.
Since (1) is ill-posed, regularization methods are used to obtain stable approximate solutions [2, 3]. Iterative regularization methods are one such class of regularization methods [4–8].In [4], Bakushinskii proposed an iterative method, namely, the iteratively regularized Gauss-Newton method, in which the iterations are defined by
(3)xk+1δ=xkδ-(αkI+Ak,δ*Ak,δ)-1Ak,δ*×[F(xkδ)-yδ+αk(xkδ-x0)],x0δ=x0,
where Ak,δ:=F′(xkδ) (here and in the following F′ denotes the Fréchet derivative of F) and (αk) is a sequence of real numbers satisfying
(4)αk>0,1≤αkαk+1≤μ1,limk→0αk=0
for some constant μ1>1. For convergence analysis, Bakushinskii used the following Hölder-type source condition on the exact solution x^ of (1)
(5)x0-x^=A*Aw,A:=F′(x^),
for some w∈X. Later Hohage [9, 10] and Langer and Hohage [11] considered the iteratively regularized Gauss-Newton method under different source conditions and stopping rules.In [5], Bakushinskii generalized the procedure in [4] by considering a generalized form of the regularized Gauss-Newton method in which the iterations are defined by
(6)xk+1δ=x0-gαk(Ak,δ*Ak,δ)Ak,δ*×[(F(xkδ)-yδ)-Ak,δ(xkδ-x0)],x0δ=x0,
where Ak,δ and (αk) are as in (3), and each gα for α>0 is a piecewise continuous function. It should be noted that the convergence of (3) was also shown by Bakushinsky and Smirnova in [12]. In [6], Blaschke et al. considered the above generalized procedure under a stopping index kδ such that
(7)∥F(xkδδ)-yδ∥≤cδ<∥F(xkδ)-yδ∥,k=0,1,…,kδ-1,c>1,
and the error estimate is obtained under the following Hölder-type source condition:
(8)x0-x^=(A*A)νw,0<ν≤1.Recently, Mahale and Nair [8], considered the iteration procedure:
(9)xk+1δ=x0-gαk(A0*A0)A0*×[(F(xkδ)-yδ)-A0(xkδ-x0)],x0δ=x0,
where A0:=F′(x0), (αk) is as in (3), and gα for α>0 is a positive real-valued piecewise continuous function defined in [0,M] with M≥∥A0∥2. In [8], the stopping index kδ for the iteration is chosen such that
(10)max{∥F(xkδ-1δ)-yδ∥,βkδ}≤τδ<max{∥F(xk-1δ)-yδ∥,βk}1≤k<kδ, where τ>0 is a sufficiently large constant not depending on δ, and
(11)βk:=∥F(xk-1δ)-yδ+A0(xkδ-xk-1δ)∥.To prove the results in [8], Mahale and Nair considered the following general source condition:
(12)x0-x^=[φ(A0*A0)]1/2w,
for some w∈X with ∥w∥≤ρ, ρ>0. Here, φ:(0,M]→(0,∞) is a continuous, strictly monotonically increasing function satisfying limλ→0φ(λ)=0 with M≥∥A0∥2.Note that the source conditions (5) and (8) involves the Fréchet derivative at the exact solution x^ which is unknown in practice. But the source condition (12) depends on the Fréchet derivative of F at x0.In [7], Kaltenbacher considered the following iteration procedure:
(13)xn+1,αδ=ProjD(F)[xn,αδ-(An*An+αI)-1ProjD(F)cc×(An*(F(xn,αδ)-yδ)+α(xn,αδ-x0))(An*An+αI)-1],
where An:=F′(xn,αδ), x0=ProjD(F)x0,αδ and proved that the sequence (xn,αδ) converges to the critical point xαδ of the Tikhonov functional, characterized by F′(xαδ)*(F(xαδ)-yδ)+α(xαδ-x0)=0. In order to obtain an error estimate of ∥xαδ-x^∥ in [7], the following two kinds of conditions are used:(a)
F′(x-)=R(x-,x)F′(x), ∥R(x-,x)-I∥≤CR∥x--x∥κ for x,x-∈Bρ(x0),(b)
F′(x-)=F′(x)R(x-,x), ∥R(x-,x)-I∥≤CR∥x--x∥κ for x,x-∈Bρ(x0), κ∈[0,1],with
(14)(1+CR(2ρ)κ)2(CR2)(2ρ)κ<1
for condition (a) and
(15)2CRρ<1
for condition (b).In [13], the author considered a particular case of the method (9), that is,
(16)xn+1,αδ=xn,αδ-(A0*A0+αI)-1×[A0*(F(xn,αδ)-yδ)+α(xn,αδ-x0)],x0,αδ=x0,
or equivalently,
(17)xn+1,αδ=x0-(A0*A0+αI)-1A0*×[(F(xn,αδ)-yδ)-A0(xn,αδ-x0)],x0,αδ=x0,
for approximately solving (1). Analysis in [13], was carried out using a suitably constructed majorizing sequence, and the stopping rule in [13] was based on this majorizing sequence.Recall [14, Definition 1.3.11], that a nonnegative increasing sequence (tn) (i.e., tn+1-tn≥0) is said to be a majorizing sequence of a sequence (xn) in X, if
(18)∥xn+1-xn∥≤tn+1-tn,∀n≥0.The majorizing sequence in [13], depends on the initial guess x0, and the conditions on x0 (see, e.g., (3.2) in [13]) are restrictive, so the method is not suitable for practical consideration.In this paper, we consider the sequence (17) and analyze it by considering its even and odd terms separately and obtain the optimal order of the error. The regularization parameter α is chosen according to the balancing principle considered by Pereverzev and Schock in [15].The organization of this paper is as follows. Proposed method and its convergence analysis are given in Section2. Error analysis and parameter choice strategy are discussed in Section 3. Section 4 deals with the implementation of the method; a numerical example is given in Section 5, and finally, the paper ends with a conclusion in Section 6.


## 2. Convergence of the Method (17)

Let(19)ynδ:=x0-Rα-1A0*[F(xnδ)-yδ-A0(xnδ-x0)],(20)xn+1δ:=x0-Rα-1A0*[F(ynδ)-yδ-A0(ynδ-x0)],
where Rα:=A0*A0+αI and x0 is the initial guess.Remark 1.
It can be seen thatynδ=x2n-1,αδ and xn+1δ=x2n,αδ, where xn,αδ is defined as in (17).Assumption 2.
There exists a constantk0>0 such that for every x,u∈D(F) and v∈X, there exists an element Φ(x,u,v)∈X satisfying
(21)[F′(x)-F′(u)]v=F′(u)Φ(x,u,v),∥Φ(x,u,v)∥≤k0∥v∥∥x-u∥
for all x,u∈D(F) and v∈X.Let(22)enδ≔∥ynδ-xnδ∥,∀n≥0.Hereafter, for convenience, we use the notationxn, yn, and en for xnδ, ynδ, and enδ, respectively.Letδ0<α0/4k0, 0<ρ≤(1/k0)[(3/2)-(2k0δ0/α0)-1], the parameter α is selected from some finite set
(23)DM:={αi:0<α0<α1<⋯<αM},γρ≔δ0α0+k02ρ2+ρ.
Throughout this paper, we assume that the operator F is Fréchet differentiable at all x∈D(F).Remark 3.
Note that if∥x0-x^∥≤ρ, then by Assumption 2, we have
(24)e0≤γρ.
This can be seen as follows:
(25)e0=∥y0-x0∥=∥Rα(x0)-1A0*(yδ-F(x0))∥=∥c-A0(x^-x0)+A0(x^-x0))c-A0(x^-x0)+A0(x^-x0)yδ-y+F(x^)-F(x0))Rα(x0)-1A0*(yδ-y+F(x^)-F(x0)Rα(x0)-1A0*cyδc-A0(x^-x0)+A0(x^-x0))Rα(x0)-1A0*∥≤∥Rα(x0)-1A0*(yδ-y)∥+∥Rα(x0)-1A0*∫01‍[F′(x0+t(x^-x0))-A0]dt(x^-x0)∥+∥Rα(x0)-1A0*A0(x^-x0)∥≤δα+k02ρ2+ρ≤δ0α0+k02ρ2+ρ=γρ≤14k0.Using the inequality (24), we prove the following.Theorem 4.
Let∥x0-x^∥≤ρ, q=k0r, where r∈((1-1-4k0γρ)/2k0,(1+(1-4k0γρ))/2k0). Let en be as in (22), and let xn and yn be as in (19) and (20), respectively, with δ∈[0,δ0) and α∈DM. Then, we have the following:(a)
∥xn-yn-1∥≤q∥yn-1-xn-1∥,(b)
∥yn-xn∥≤q2∥yn-1-xn-1∥,(c)
en≤q2nγρ,(d)
xn,yn∈Br(x0):={x∈X:∥x-x0∥<r}.Proof.
Observe that ifxn,yn∈Br(x0), for all n≥0, then by Assumption 2, we have
(26)xn-yn-1=-Rα(x0)-1A0*[F(yn-1)-F(xn-1)-A0(yn-1-xn-1)]=Rα(x0)-1A0*×∫01‍[F′(yn-1+t(xn-1-yn-1))-A0]×(xn-1-yn-1)dt=Rα(x0)-1A0*A0×∫01‍Φ(x0,yn-1+t(xn-1-yn-1),xn-1-yn-1)dt,
and hence
(27)∥xn-yn-1∥≤k0r∥xn-1-yn-1∥.
This proves (a).
Again observe that ifxn,yn∈Br(x0), for all n≥0,
(28)yn-xn=-Rα(x0)-1A0*×[(F(xn)-F(yn-1))-A0(xn-yn-1)]=-Rα(x0)-1A0*×∫01‍[F′(yn-1+t(xn-yn-1))-A0]×dt(xn-yn-1)=-Rα(x0)-1A0*A0×∫01‍Φ(x0,yn-1+t(xn-yn-1),xn-yn-1)dt,
and hence by Assumption 2 and (27), we have
(29)∥yn-xn∥≤k0r∥xn-yn-1∥≤q2∥xn-1-yn-1∥.
This proves (b).
Thus, ifxn,yn∈Br(x0), for all n≥0, then (c) follows from (a) and (b). Now, we will prove using induction that xn,yn∈Br(x0), for all  n≥0. Note that x0,y0∈Br(x0), and hence by (27) and Remark 3,
(30)∥x1-x0∥≤∥x1-y0∥+∥y0-x0∥≤(1+q)e0≤γρ1-q≤r,
that is, x1∈Br(x0), again by (29) and Remark 3,
(31)∥y1-x0∥≤∥y1-x1∥+∥x1-x0∥≤q2e0+(1+q)e0≤γρ1-q≤r,
that is, y1∈Br(x0). Suppose that xk,yk∈Br(x0) for some k>1. Then, since
(32)∥xk+1-x0∥≤∥xk+1-xk∥+∥xk-xk-1∥+⋯+∥x1-x0∥,
we shall first find an estimate for ∥xk+1-xk∥. Note that by (a), (b), and (c), we have
(33)∥xk+1-xk∥≤∥xk+1-yk∥+∥yk-xk∥≤(q+1)∥yk-xk∥≤(1+q)q2ke0.
Therefore by (32) and Remark 3, we have
(34)∥xk+1-x0∥≤(1+q)[q2k+q2(k-1)+⋯+1]e0≤(1+q)1-q2k+11-q2e0≤γρ1-q≤r,
that is, xk+1∈Br(x0). So by induction xn∈Br(x0) for all n≥0. Again by (a), (b) and (34) we have
(35)∥yk+1-x0∥≤∥yk+1-xk+1∥+∥xk+1-x0∥≤q2k+2e0+(1+q)[q2k+q2(k-1)+⋯+1]e0≤(1+q)1-q2k+31-q2e0≤γρ1-q≤r.
Thus, yk+1∈Br(x0), and hence by induction, yn∈Br(x0) for all n≥0. This completes the proof of the theorem.The main result of Section2 is the following.Theorem 5.
Letyn and xn be as in (19) and (20), respectively, with δ∈(0,δ0], and assumptions of Theorem 4 hold. Then, (xn) is a Cauchy sequence in Br(x0) and converges to xαδ∈Br(x0)-. Further, A0*(F(xαδ)-yδ)+α(xαδ-x0)=0, and
(36)∥xn-xαδ∥≤q2nγρ(1-q).Proof.
Using the relation (33), we obtain
(37)∥xn+m-xn∥≤∑i=0i=m-1‍∥xn+i+1-xn+i∥≤∑i=0i=m-1‍(1+q)q2(n+i)e0≤(1+q)(q2n-q2n+2m)1-q2e0≤q2n1-qγρ.
Thus, xn is a Cauchy sequence in Br(x0), and hence it converges, say to xαδ∈Br(x0)-. Observe that ∥xn+1-yn∥≤k0r∥xn-yn∥ and k0r=q<1. Hence, (yn) also converges to xαδ.
Now, byn→∞ in (20), we obtain xαδ=x0-(A0*A0+αI)-1A0*[F(xαδ)-yδ-A0(xαδ-x0)], that is,
(38)A0*(F(xαδ)-yδ)+α(xαδ-x0)=0.
This completes the proof.


## 3. Error Analysis

The next assumption on source condition is based on a source functionφ and a property of the source function φ. We will be using this assumption to obtain an error estimate for ∥xαδ-x^∥.Assumption 6.
There exists a continuous, strictly monotonically increasing functionφ:(0,a]→(0,∞) with a≥∥F′(x0)∥2 satisfying(i)
limλ→0φ(λ)=0,(ii)
supλ≥0(αφ(λ)/(λ+α))≤φ(α), for all λ∈(0,a],(iii)
there existsv∈X such that
(39)x0-x^=φ(A0*A0)v.Theorem 7.
Letxαδ be as in (38). Then,
(40)∥xαδ-x^∥≤11-q(δα+φ(α)).Proof.
LetM=∫01F′(x^+t(xαδ-x^))dt. Then,
(41)F(xαδ)-F(x^)=M(xαδ-x^),
and hence by (38), we have (A0*M+αI)(xαδ-x^)=A0*(yδ-y)+α(x0-x^). Thus,
(42)xαδ-x^=(A0*A0+αI)-1×[A0*(yδ-y)+α(x0-x^)+A0*(A0-M)(xαδ-x^)]=s1+s2+s3,
where s1:=(A0*A0+αI)-1A0*(yδ-y), s2:=(A0*A0+αI)-1α(x0-x^), and s3:=(A0*A0+αI)-1A0*(A0-M)(xαδ-x^). Note that
(43)∥s1∥≤δα,
by Assumption 6(44)∥s2∥≤φ(α)
and by Assumption 2(45)∥s3∥≤k0r∥xαδ-x^∥.
The result now follows from (42), (43), (44), and (45). This completes the proof of the theorem.


### 3.1. Error Bounds under Source Conditions

Combining the estimates in Theorems5 and 7, we obtain the following.Theorem 8.
Letxnδ be defined as in (20). If all assumptions of Theorems 5 and 7 are fulfilled, then
(46)∥xnδ-x^∥≤q2n1-qγρ+11-q(δα+φ(α)).
Further, if nδ:=min{n:q2n<δ/α}, then
(47)∥xnδδ-x^∥≤C~(δα+φ(α)),
where C~:=(1/(1-q))(γρ+1).


### 3.2. A Priori Choice of the Parameter

Observe that the upper bound(δ/α)+φ(α) in Theorem 8 is of optimal order for the choice α:=αδ which satisfies δ/αδ=φ(α). Now, using the function ψ(λ):=λφ-1(λ), 0<λ≤a, we have δ=αφ(α)=ψ(φ(α)) so that αδ=φ-1[ψ-1(δ)]. Here, φ-1 means the inverse function of φ.Theorem 9.
Suppose that all assumptions of Theorems5 and 7 are fulfilled. For δ>0, let αδ=φ-1[ψ-1(δ)], and let nδ be as in Theorem 8 with α=αδ. Then,
(48)∥xnδδ-x^∥=O(ψ-1(δ)).


### 3.3. Adaptive Choice of the Parameter

In the balancing principle considered by Pereverzev and Schock in [15], the regularization parameter α=αi is selected from some finite set
(49)DN≔{αi:0<α0<α1<⋯<αN}.
Let
(50)ni=min{n:q2n≤δαi},
and let xni,αiδ:=xniδ, where xniδ is defined as in (20) with α=αi and n+1=ni. Then, from Theorem 8, we have
(51)∥xni,αiδ-x^∥≤C~(δαi+φ(αi)),∀i=1,2,…,N.
Precisely, we choose the regularization parameter α=αk from the set DN defined by
(52)DN:={αi=μiα0,i=1,2,…,N},
where μ>1.To obtain a conclusion from this parameter choice, we consider all possible functionsφ satisfying Assumptions 2 and φ(αi)≤δ/αi. Any of such functions is called admissible for x^, and it can be used as a measure for the convergence of xαδ→x^ (see [16]).The main result of Section3 is the following.Theorem 10.
Assume that there existsi∈{0,1,…,N} such that φ(αi)≤δ/αi. Let the assumptions of Theorems 5 and 7 be fulfilled, and let
(53)l:=max{i:φ(αi)≤δαi}<N,k=max{i:∀j=1,2,…,i;∥xni,αiδ-xnj,αjδ∥≤4C~δαj},
where C~ is as in Theorem 8. Then, l≤k and
(54)∥xnk,αkδ-x^∥≤6C~μψ-1(δ).Proof.
The proof is analogous to the proof of Theorem  4.4 in [13] and is omitted therefore here.


## 3.1. Error Bounds under Source Conditions

Combining the estimates in Theorems5 and 7, we obtain the following.Theorem 8.
Letxnδ be defined as in (20). If all assumptions of Theorems 5 and 7 are fulfilled, then
(46)∥xnδ-x^∥≤q2n1-qγρ+11-q(δα+φ(α)).
Further, if nδ:=min{n:q2n<δ/α}, then
(47)∥xnδδ-x^∥≤C~(δα+φ(α)),
where C~:=(1/(1-q))(γρ+1).


## 3.2. A Priori Choice of the Parameter

Observe that the upper bound(δ/α)+φ(α) in Theorem 8 is of optimal order for the choice α:=αδ which satisfies δ/αδ=φ(α). Now, using the function ψ(λ):=λφ-1(λ), 0<λ≤a, we have δ=αφ(α)=ψ(φ(α)) so that αδ=φ-1[ψ-1(δ)]. Here, φ-1 means the inverse function of φ.Theorem 9.
Suppose that all assumptions of Theorems5 and 7 are fulfilled. For δ>0, let αδ=φ-1[ψ-1(δ)], and let nδ be as in Theorem 8 with α=αδ. Then,
(48)∥xnδδ-x^∥=O(ψ-1(δ)).


## 3.3. Adaptive Choice of the Parameter

In the balancing principle considered by Pereverzev and Schock in [15], the regularization parameter α=αi is selected from some finite set
(49)DN≔{αi:0<α0<α1<⋯<αN}.
Let
(50)ni=min{n:q2n≤δαi},
and let xni,αiδ:=xniδ, where xniδ is defined as in (20) with α=αi and n+1=ni. Then, from Theorem 8, we have
(51)∥xni,αiδ-x^∥≤C~(δαi+φ(αi)),∀i=1,2,…,N.
Precisely, we choose the regularization parameter α=αk from the set DN defined by
(52)DN:={αi=μiα0,i=1,2,…,N},
where μ>1.To obtain a conclusion from this parameter choice, we consider all possible functionsφ satisfying Assumptions 2 and φ(αi)≤δ/αi. Any of such functions is called admissible for x^, and it can be used as a measure for the convergence of xαδ→x^ (see [16]).The main result of Section3 is the following.Theorem 10.
Assume that there existsi∈{0,1,…,N} such that φ(αi)≤δ/αi. Let the assumptions of Theorems 5 and 7 be fulfilled, and let
(53)l:=max{i:φ(αi)≤δαi}<N,k=max{i:∀j=1,2,…,i;∥xni,αiδ-xnj,αjδ∥≤4C~δαj},
where C~ is as in Theorem 8. Then, l≤k and
(54)∥xnk,αkδ-x^∥≤6C~μψ-1(δ).Proof.
The proof is analogous to the proof of Theorem  4.4 in [13] and is omitted therefore here.


## 4. Implementation of the Method

Finally, the balancing algorithm associated with the choice of the parameter specified in Theorem10 involves the following steps.(i)
Chooseα0>0 such that δ0<α0/4k0 and μ>1.(ii)
ChooseN big enough but not too large and αi:=μiα0, i=0,1,2,…,N.(iii)
Chooseρ≤((3/2)-(2k0δ0/α0)-1)/k0.


### 4.1. Algorithm

(1)
Seti=0.
(2)
Chooseni=min{n:q2n≤δ/αi}.
(3)
Solvexni,αiδ=xniδ by using the iteration in (19) and (20) with n=ni and α=αi.
(4)
If∥xαiδ-xαjδ∥>4C~(δ/αj),j<i, then take k=i-1 and return xαkδ.
(5)
Else seti=i+1, and return to step (2).


## 4.1. Algorithm

(1)
Seti=0.
(2)
Chooseni=min{n:q2n≤δ/αi}.
(3)
Solvexni,αiδ=xniδ by using the iteration in (19) and (20) with n=ni and α=αi.
(4)
If∥xαiδ-xαjδ∥>4C~(δ/αj),j<i, then take k=i-1 and return xαkδ.
(5)
Else seti=i+1, and return to step (2).


## 5. Numerical Example

We apply the algorithm by choosing a sequence of finite dimensional subspace(Vn) of X with dim Vn=n+1. Precisely, we choose Vn as the linear span of {v1,v2,…,vn+1}, where vi, i=1,2,…,n+1 are the linear splines in a uniform grid of n+1 points in [0,1].We consider the same example of nonlinear integral operator as in [17, Section 4.3]. Let F:D(F)⊆L2(0,1)→L2(0,1) be defined by
(55)F(u):=∫01‍k(t,s)u3(s)ds,
where
(56)k(t,s)={(1-t)s,0≤s≤t≤1(1-s)t,0≤t≤s≤1.
The Fréchet derivative of F is given by
(57)F′(u)w=3∫01‍k(t,s)(u(s))2w(s)ds.Note that foru,v>0,
(58)(F′(v)-F′(u))w=(3∫01‍k(t,s)(u(s))2ds)×[∫01k(t,s)((v(s))2-(u(s))2)w(s)ds∫01‍k(t,s)(u(s))2ds]:=F′(u)Φ(v,u,w),
where Φ(v,u,w)=(∫01k(t,s)((v(s))2-(u(s))2)w(s)ds)/(∫01k(t,s)(u(s))2ds).Observe that(59)Φ(v,u,w)=∫01‍k(t,s)((v(s))2-((u(s))2)w(s)ds∫01‍k(t,s)(u(s))2ds=∫01‍k(t,s)(u(s)+v(s))(v(s)-u(s))w(s)ds∫01‍k(t,s)(u(s))2ds.
So, Assumption 2 is satisfied with k0≥∥(∫01k(t,s)(u(s)+v(s))ds)‍/(∫01k(t,s)(u(s))2ds)∥.In our computation, we takey(t)=(t-t11)/110 and yδ=y+δ. Then, the exact solution
(60)x^(t)=t3.
We use
(61)x0(t)=t3+356(t-t8)
as our initial guess, so that the function x0-x^ satisfies the source condition
(62)x0-x^=φ(F′(x0))(x^x0)2,
where φ(λ)=λ.Observe that while performing numerical computation on finite dimensional subspace(Vn) of X, one has to consider the operator PnF′(·)Pn instead of F′(·), where Pn is the orthogonal projection on to Vn. Thus, incurs an additional error ∥PnF′(·)Pn-F′(·)∥=O(∥F′(·)(I-Pn)∥).Let∥F′(·)(I-Pn)∥≤εn. For the operator F′(·) defined in (57), εn=O(n-2) (cf. [18]). Thus, we expect to obtain the rate of convergence O((δ+n-2)1/2).We chooseα0=1.7×(δ+εn), μ=1.7, k0=1, and q=0.71. The results of the computation are presented in Table 1. The plots of the exact solution and the approximate solution obtained are given in Figures 1 and 2.Table 1
Iterations and corresponding error estimates.
n
k
n
k
δ
+
ε
n
α
k
∥
x
n
k
,
α
k
δ
-
x
^
∥
∥
x
n
k
,
α
k
δ
-
x
^
∥
/
(
δ
+
ε
n
)
1
/
2
8
2
6
0.1016
0.8483
0.2622
0.8228
16
2
6
0.1004
0.8385
0.1958
0.6179
32
2
6
0.1001
0.8360
0.1438
0.4545
64
2
6
0.1000
0.8354
0.1056
0.3338
128
2
6
0.1000
0.8353
0.0789
0.2495
256
2
6
0.1000
0.8352
0.0661
0.1948
512
2
6
0.1000
0.8352
0.0522
0.1652
1024
2
6
0.1000
0.8352
0.0459
0.1450Curves of the exact and approximate solutions.
(a)
(b)
(c)
(d)Curves of the exact and approximate solutions.
(a)
(b)
(c)
(d)


## 6. Conclusion

In this paper, we considered a modified Gauss-Newton method for approximately solving the nonlinear ill-posed operator equationF(x)=y, where F:D(F)⊆X→Y is a nonlinear operator between the Hilbert spaces X and Y. The same method was considered in [13] by the author, but the analysis in [13] was based on a suitably constructed majorizing sequence. In this paper, we analyze the sequence by considering its even and odd terms separately. The analysis in this paper is easier than that of [13]. We use the adaptive method considered by Pereverzev and Schock in [15] for choosing the regularization parameter. The optimality of this method is proved under a general source condition. Finally, a numerical example of nonlinear integral equation shows the performance of this method.



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*Source: 439316-2013-01-30.xml*