\(^\ast \)Department of Mathematicsal Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: ioannisa@cameron.edu.

\(^\S \)Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, India-757 025, e-mail:sgeorge@nitk.ac.in.

1 Introduction

In this study we are concerned with the problem of approximating a solution of non-linear ill-posed equation

where \(A\) is a nonlinear operator defined on a subset \(D=D(A)\) of a Hilbert space \(X\), and with range \(R(A)\) in a Hilbert space \(Y.\) Equation (1.1) is ill-posed in the sense that the solution of (1.1) does not depend continuously on the data \(y.\) Many regularization methods such as Tikhonov regularization [ 9 , 10 , 23 , 25 , 27 , 32 ] , Gauss-Newton method [ 6 ] and other methods [ 21 ] , [ 22 ] have been used to approximate solution of equation (1.1).

Many problems from computational sciences and other disciplines can be brought in a form similar to equation (1.1) using mathematical modelling [ 1 ] , [ 5 ] , [ 7 ] , [ 8 ] , [ 29 ] , [ 30 ] . The solutions of these equations can rarely be found in closed form. That is why most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure, while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls.

George and collaborators (see [ 13 ] – [ 20 ] ) solved equation (1.1) for the special case when \(A\) is a Hammerstein-type operator. A Hammerstein-type operator is of the form \(A=MF,\) where \(F:D(F)\subset X\rightarrow Z\) is a nonlinear and \(M:Z\rightarrow Y\) is a bounded linear operator with \(X,Y\) and \(Z\) are Hilbert spaces. Hence, (1.1) becomes

In particular George and Kunhanandan [ 16 ] assumed that a solution \(x^\ast \in D(F)\) of (1.2) satisfies

and \(y^\delta \in Y\) are the available noisy data such that

Then, for fixed \(\alpha {\gt} 0, \delta {\gt}0\), the Newton-Tikhonov (NT) method defined by

where \(z_\alpha ^\delta \) is an approximation of the solution of the equation \(M(z)=y^\delta \) (see Section 5) was used to generate a sequence \(\{ x_{n,\alpha }^\delta \} \) converging quadratically to a solution \(x_\alpha ^\delta \) of the equation

provided that certain Kantrovich-type criteria are satisfied.

In the present paper we expand the applicability of (NT) by using more precise majorizing sequence for \(\{ x_{n,\alpha }^\delta \} \) than the ones given in [ 17 ] . This way we provide a semilocal convergence analysis for (NT) with the following advantages over the work in [ 16 ] under the same computational cost:

1. Weaker sufficient convergence criteria;

2. Tighter error estimates on the distances \(\| x_{n+1,\alpha }^\delta -x_{n,\alpha }^\delta \| \) and

3. An at least as precise information on the location of the solution.

These advantages are obtained, since we use the more precise center-Lipschitz condition instead of the Lipschitz condition for the computation of the upper bounds on the norms \(\| F'(x_{n,\alpha }^\delta )^{-1}\| \) (see (C3) and (C4) in Section 3). We also study the semilocal convergence of the simplified Newton-Tikhonov method (SNT) defined by

(SNT) method is used as predictor for (NT) method since the former converges under weaker sufficient convergence criteria than the latter.

The paper is organized as follows. Section 2 contains results on scalar sequences that are majorizing for (NT). Sections 3 and 4 contain respectively, the semilocal convergence of (NT) and (SNT). The applications are given in the concluding Section 5.

2 Majorizing Sequences

We present auxiliary results on scalar sequences which shall be shown to be majorizing for \(\{ x_{n,\alpha }^\delta \} \) in Section 3.

Then, using a simple inductive argument we obtain the following result where we compare the three scalar sequences. We also have the following generalization of Lemma 2.6.

3 Semilocal Convergence of (NT)

We present the semi-local convergence of \(\{ x_{n,\alpha }^\delta \} \) in this section. Let \(U(x,r)\) and \(\overline{U(x,r)}\) stand, respectively, for the open and closed ball in \(X\) with center \(x\) and radius \(r{\gt}0.\) Let \(L(X,Y)\) stand for the space of bounded linear operators from \(X\) into \(Y.\) We assume throughout this section that the following \((C)\) conditions hold:

1. \(F:D(F)\subseteq X\rightarrow Y\) is Fr\(\grave{e}\)chet differentiable

2. There exists \(x_0\in D(F)\) such that \(F'(x_0)^{-1}\in L(Y,X)\) and

   \[ \| F'(x_0)^{-1}\| \leq b \]

3. There exists \(L_0 {\gt}0\) such that \(F'\) satisfies the center-Lipschitz conditions

   \[ \| F'(x)-F'(x_0)\| \leq L_0\| x-x_0\| \]

   holds for all \(x\in D(F).\)

4. There exists \(L{\gt} 0\) such that \(F'\) satisfies the Lipschitz condition

   \[ \| F'(x)-F'(y)\| \leq L\| x-y\| \]

   holds for all \(x\) and \(y\) in \(D(F).\)

5. \(\| F'(x_0)^{-1}(F(x_0)-z_\alpha ^\delta )\| \leq r\)

6. \(h_2=bL_2r \leq 1,\) where \(L_2=\tfrac {1}{4}(4L_0+\sqrt{3L_0L+8L_0^2}+\sqrt{3L_0L})\) and

7. \(\overline{U(x,r^\ast )}\subseteq D(F),\) where \(r^\ast \) is defined in (2.12).

We present the following semi-local convergence result for \(\{ x_{n,\alpha }^\delta \} .\)

We use mathematical induction to prove that

and

for each \(k=0,1,2,\ldots .\) Let \(v\in \overline{U(x_{1,\alpha }^\delta , r^\ast -r_{1,\alpha }^\delta )}.\) Then, we obtain

which implies \(v\in \overline{U(x_{0,\alpha }^\delta , r^\ast -r_{0,\alpha }^\delta )}.\) Note also that

Hence, estimates (3.26) and (3.27) hold for \(k =0.\) Suppose that these estimates hold for \(n\leq k.\) Then, we have that

and

for each \(\theta \in [0,1].\)

Using (C3), Lemmas 2.5 (see also Lemma 2.1 in [ 3 ] ) and the induction hypotheses we get

It follows from (3.28) and the Banach Lemma on invertible operators [ 1 ] , [ 5 ] that \(F'(x_{k+1,\alpha }^\delta )^{-1} \in L(Y,X)\) and

Using (NT) we obtain the approximation

In view of (C4), (2.8), (3.30), (C3) for \(k=0\) and the induction hypotheses we get

Moreover, by (NT), (2.8), (3.29) and (3.31) we get that

which completes the induction for (3.26). Furthermore, let
\(w\in \overline{U(x_{k+2,\alpha }^\delta , r^\ast -r_{k+2,\alpha }^\delta )}.\) Then, we have that

That is \(w\in \overline{U(x_{k+1,\alpha }^\delta , r^\ast -r_{k+1,\alpha }^\delta )}.\) Lemma 2.6 implies that \(\{ r_{k,\alpha }^\delta \} \) is a complete sequence. It then follows from (3.26) and (3.27) that \(\{ x_{k,\alpha }^\delta \} \) is also complete sequence in the Hilbert space \(X\) and as such it converges to some \(x_\alpha ^\delta \in \overline{U(x_0, r^\ast )}\) (since \(\overline{U(x_0, r^\ast )}\) is a closed set). By letting \(k\rightarrow \infty \) in (3.31) we obtain \(F(x_\alpha ^\delta )=z_\alpha ^\delta .\) Estimate (3.25) is obtained from (3.24) by using standard majorization techniques [ 1 ] , [ 5 ] , [ 15 ] . The proof of the Theorem is complete.

The rest of the results in [ 16 ] can be improved along the same lines by simply using, respectively, \(\bar{r}^\ast , L_0\) instead of \(\bar{t}^\ast , L.\) In order for us to make the paper is self contained as possible we present the proof of one of them and for the proof of the rest we refer the reader to [ 16 ] . Using (C3) (instead of (C4)) used in [ 16 ] , we get that

which shows (3.34). The proof of the proposition is complete.

The following is a consequence of Theorem 3.1 and Proposition 3.3.

4 Semilocal convergence of (SNT)

We present the semilocal convergence of (SNT).

Let us define operator \(T\) on \( \overline{U(x_0, \rho )}\) by

Then, we shall show \(T\) is a contraction on \( \overline{U(x_0, \rho )}\) and maps \( \overline{U(x_0, \rho )}\) into itself. Indeed, we have that for \( x,y\in \overline{U(x_0, \rho )}\)

and

But, we have \(bL_0\rho {\lt}1.\) Hence, \( T\) is a contraction operator. Let \( x\in \overline{U(x_0, \rho )}.\) Then, we have that

and

by the choice of \(\rho .\) The proof of Theorem is complete.

5 Applications

Let us consider the nonlinear Hammerstein operator equation (c.f. [ 28 ] )

where \(m\) is continuous and \(p\) is differentiable with respect to the second variable. Define \(F:D(F)=H^1(]0,1[)\rightarrow L^2(]0,1[)\) by

and \(M:L^2(]0,1[) \rightarrow L^2(]0,1[)\) by

Then, \(F\) is Fr\(\grave{e}\)chet differentiable and we have that

If in addition \(M_1:H^1(]0,1[)\mapsto H^1(]0,1[)\) is defined by \((M_1x)(t):=\partial _2p(t,x(t))\) is locally Lipschitz continuous, one can compute the required constants \(L_0\) and \(L.\) If we further assume the existence of a constant \(\kappa _1 {\gt}0\) such that \(\partial _2p(t,x(t)) \geq \kappa _1\) for all \(t\in [0,1]\) and \(x(t) \in U(x_0, r^\ast ),\) then \(F'(x)^{-1}\) exists and is bounded operator.

Equation (1.2) is equivalent to

where \(x_0,\) is an initial guess. Therefore the solution x of (1.2) is obtained by first solving

for \(z\) and then solving

for \(x\in D(F).\) Let \(\alpha {\gt} 0,\) \(\delta {\gt}0\) be fixed. Then, we consider the regularized solution of (5.36) with \(y^\delta \) in place of y as

if case \(M\) in (5.36) is positive self adjoint and \(Z=Y.\) Otherwise, we set

Note that (5.38) is the simplified or Lavrentiev regularization of equation (5.36) and (5.39) is the Tikhonov regularization of (5.36).

With these choices of operators the rest of the results in [ 16 ] involving (SNT) type methods (i.e., using \( L_0\) instead of \(L\)) can be improved.

In view of the estimate in the Corollory 3.4, the next task is to find an estimate \(\| F(x^\ast )-z_\alpha ^\delta \| .\) For this, let us introduce the notation;

We may observe that

and

Note that for \(u \in R(M^*M)\) with \(u = M^*Mz\) for some \(z \in Z,\)

as \(\alpha \rightarrow 0.\) Now since \( \| \alpha (M^*M + \alpha I)^{-1}\| \leq 1\) for all \(\alpha {\gt} 0,\) it follows that for every \(u \in \overline{R(M^\ast M)},\) \(\| \alpha (M^*M + \alpha I)^{-1}u\| \rightarrow 0\) as \(\alpha \rightarrow 0.\) Thus we have the following theorem.

5.1 Error bounds under source conditions

In view of the above theorem, we assume that

for some positive monotonic increasing function \(\varphi \) defined on \((0, \| M\| ^2]\) such that

Suppose \(\varphi \) is a source function in the sense that \(x^\ast \) satisfies a source condition of the form

such that

then the assumption (5.42) is satisfied. Note that if \(F(x^\ast \! )-\! F(x_0)\! \in \! {R((M^\ast \! M)^\nu \! )},\) for some \(\nu \) with, \(0 {\lt} \nu \leq 1,\) then by (5.41)

Thus in this case \(\varphi (\lambda ) = \tfrac {\lambda ^\nu }{\| \omega \| }\) satisfies the assumption (5.42). Therefore by (5.40) and by the assumption (5.42), we have

So, we have the following theorem.

5.2 A priori choice of the parameter

Note that the estimate \(\varphi (\alpha ) + \tfrac {\delta }{\sqrt{\alpha }}\) in (5.43) attains minimum for the choice \(\alpha :=\alpha _\delta \) which satisfies \(\varphi (\alpha _\delta ) = \tfrac {\delta }{\sqrt{\alpha _\delta }}.\) Let \(\psi (\lambda ) := \lambda \sqrt{\varphi ^{-1}(\lambda )}, 0 {\lt} \lambda \leq \| M\| ^2.\) Then we have \(\delta = \sqrt{\alpha _\delta }\varphi (\alpha _\delta ) = \psi (\varphi (\alpha _\delta )), \) and

So the relation (5.44) leads to

Theorem 5.5 and the above observation leads to the following.

5.3 An adaptive choice of the parameter

The error estimate in the above Theorem has optimal order with respect to \(\delta .\) Unfortunately, an a priori parameter choice (5.45) cannot be used in practice since the smoothness properties of the unknown solution \(\hat{x}\) reflected in the function \(\varphi \) are generally unknown. There exist many parameter choice strategies in the literature, for example see [ 6 ] , [ 11 ] , [ 12 ] , [ 18 ] , [ 19 ] , [ 31 ] and [ 33 ] .

In [ 26 ] , Pereverzev and Schock considered an adaptive selection of the parameter which does not involve even the regularization method in an explicit manner. In this method the regularization parameter \(\alpha _i\) are selected from some finite set \(\{ \alpha _i: 0 {\lt} \alpha _0 {\lt} \alpha _1 {\lt} \ldots {\lt} \alpha _N\} \) and the corresponding regularized solution, say \(u_{\alpha _i}^\delta \) are studied on-line. Later George and Nair [ 20 ] considered the adaptive selection of the parameter for choosing the regularization parameter in Newton-Lavrentiev regularization method for solving Hammerstein-type operator equation. In this paper also, we consider the adaptive method for selecting the parameter \(\alpha \) in \(x_{\alpha , n}^\delta \). The rest of this section is essentially a reformulation of the adaptive method considered in [ 26 ] in a special context.

Let \(i\in \{ 0, 1, 2, \ldots , N\} \) and \(\alpha _i = \mu ^{2i}\alpha _0\) where \(\mu {\gt} 1\) and \(\alpha _0 = \delta ^2.\) Let

The proof of the next theorem is analogous to the proof of Theorem 1.2 in [ 26 ] , but for the sake of completeness, we supply its proof as well.

Note that, to prove \(l\leq k\), it is enough to prove that, for \(i=1,\ldots ,N\)

For \(j \leq i,\)

This proves the relation \(l \leq k.\) Now since \(\sqrt{\alpha _{l+m}} = \mu ^m\sqrt{\alpha _{l}},\) by using triangle inequality successively, we obtain

Therefore by (5.43) and (5.46) we have

The last step follows from the inequality \(\sqrt{\alpha _\delta }\leq \sqrt{\alpha _{l+1}}\leq \mu \sqrt{\alpha _l}\) and \(\tfrac {\delta }{\sqrt{\alpha _\delta }} = \psi ^{-1}(\delta ).\) This completes the proof.

5.4 Stopping Rule

Note that

so if

and \(r^\ast {\lt} r.\) Then hypotheses of Theorem 3.1 hold. Again since \(\alpha _j = \mu ^{2j}\delta ^2, \linebreak \tfrac {\delta }{\sqrt{\alpha _k}} = \mu ^{-k};\) the condition (5.48) with \(\alpha = \alpha _k\) takes the form

and \(r^\ast {\lt} r.\) Then, we have arrived at the algorithm which guarantees

with \(n_k=\min \{ n:r^\ast -r_{n,\alpha }^\delta \leq \mu ^{-j}, j=1,2,\ldots , i-1\} \) and improves the corresponding one in [ 16 ] .

5.5 Algorithm:

Note that for \(i,j \in \{ 0,1,2,\ldots , n\} \)

Therefore the adaptive algorithm associated with the choice of the parameter specified in the above theorem is as follows.

begin

  i=0

  repeat

    i=i+1

    Solve for \( w_i: (M^\ast M + \alpha _iI)w_i = M^\ast (y^\delta -MF(x_0))\)

    j=-1

    repeat

      j=j+1

      Solve for \(z_{i,j}: (M^\ast M + \alpha _jI)z_{i,j} =(\alpha _j-\alpha _i)w_i\)

    until (\(\| z_{i,j}\| \leq 4\mu ^{-j}\)AND \(j{\lt}i\))

  until (\(\| z_{i,j}\| \leq 4\mu ^{-j}\))

  k=i-1.

  m=0

  repeat

    m=m+1

  until (\((r^\ast -r_{m,\alpha }^\delta ) {\gt} \tfrac {1}{\mu ^k}\))

  \(n_k=m\)

  for l=1 to \(n_k\)

    Solve for \(u_{l-1}: F'(x_{l-1, \alpha _k}^\delta )u_{l-1} = F(x_{l-1, \alpha _k}^\delta )-z_{\alpha _k}^\delta \)

    \( x_{l, \alpha _k}^\delta : = x_{l-1, \alpha _k}^\delta -u_{l-1}\)

end

6 Numerical Example

In this section we consider a example for illustrating the algorithm mentioned in the above section.