ISRN Applied MathematicsVolume 2012 (2012), Article ID 783579, 22 pagesdoi:10.5402/2012/783579

Research Article

Two-Step Newton-Tikhonov Method for Hammerstein-Type Equations: Finite-Dimensional Realization

Santhosh George and Monnanda Erappa Shobha

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

Received 26 December 2011; Accepted 26 January 2012

Academic Editors: A. Bellouquid, A. El-Sayed, A. J. Kearsley, J. Míguez, and J. Shen

Copyright © 2012 Santhosh George and Monnanda Erappa Shobha. This 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.

Abstract.

Finite-dimensional realization of a Two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equations . Here is nonlinear monotone operator, is a bounded linear operator, is a real Hilbert space, and is a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operator and the second one involves the Fréchet derivative of at an initial approximation of the the solution : balancing principle of Pereverzev and Schock (2005) is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.

1. Introduction

Tikhonov’s regularization (e.g., [1]) method has been used extensively to stabilize the approximate solution of nonlinear ill-posed problems. In recent years, increased emphasis has been placed on iterative regularization procedures [2, 3] for obtaining the approximate solution of such problems. In this paper, we examine the use of iterative regularization procedures for Hammerstein-type [4, 5] equations of the form where , is nonlinear monotone operator, does not exists, and is a bounded linear operator. Throughout this paper, is the domain of is the Fréchet derivative of is a real Hilbert space, and is a Hilbert space. The inner product and the corresponding norm are denoted by and , respectively.

Recall that [6] the operator is said to be a monotone operator if , for all .

It is assumed throughout that are the available noisy data with and possesses a uniformly bounded Fréchet derivative for each (cf. [7]), that is, for some .

Observe that the solution of (1.1) with in place of can be obtained by first solving for and then solving the nonlinear problem In [4, 5, 8] this was exploited. In [4], is approximated with where and then solve (1.5) iteratively using the following Newton-type procedure: with and obtained local linear convergence. Here and below is a known initial approximation of the solution of (1.1) such that .

In [8], to solve (1.5), George and Kunhanandan used the iteration where and obtained local quadratic convergence.

A sequence in with is said to be convergent of order , if there exist positive reals and such that, for all , If a sequence satisfies , , then is said to be linearly convergent.

As in [8], it is assumed that the solution of (1.1) satisfies The regularization parameter is chosen from a finite set using the adaptive method considered by Pereverzev and Schock in [9].

In [10], Argyros and Hilout considered a method called Two-Step Directional Newton Method (TSDNM) for approximating a zero of a differentiable function defined on a convex subset of a Hilbert space with values in . Motivated by TSDNM, in [11], we propose a Two-Step Newton-Tikhonov Methods (TSNTM) for solving (1.1).

In fact, in [11] we consider two cases of , in the first case we assume that exist and in the second case we assume is monotone. In this paper we consider the finite-dimensional realization of the second case, that is, is monotone. The finite-dimensional realization of the method and associated algorithm are proposed for which local-cubic convergence is established theoretically and validated numerically.

The organization of this paper is as follows. Section 2 deals with Discretized Tikhonov regularization and Section 3 investigates the convergence of the Discretized TSNTM. Section 4 discusses the algorithm and finally the paper ends with a numerical example in Section 5.

2. Discretized Tikhonov Regularization

This section deals with discretized Tikhonov regularized solution of (1.4) and (an a priori and an a posteriori) error estimate for using an error estimate for from [8].

The following assumption is used in [8] to obtain the error estimate.

Assumption 2.1. There exists a continuous, strictly monotonically increasing function with satisfying(i), (ii)(iii)for some such that .

Remark 2.2. The functions for and for satisfy the above assumption (see [12]).

Theorem 2.3 (cf. [8], Theorem 4.3). Let be as in (1.9) and Assumption 2.1 holds. Then Let be a family of orthogonal projections on . Let and is such that ,  and . We assume that and as . The above assumption is satisfied if, pointwise and if and are compact operators. Further we assume that and where . The discretized Tikhonov regularization method for the regularized equation (1.4) consists of solving the equation

Theorem 2.4. Suppose assumptions in Theorem 2.3 hold. Let be as in (2.7) and . Then where .

Proof . Let . Then Now the result follows from (2.9), Theorem 2.3 and the following triangle inequality:

2.1. A Priori Choice of the Parameter

Note that the estimate in (2.8) is of optimal order for the choice which satisfies . Let . Then we have and So the relation (2.8) leads to .

2.2. An Adaptive Choice of the Parameter

In this subsection, we consider the balancing principle established by Pereverzev and Shock [9] for choosing the parameter . Let be the set of possible values of the parameter .

Let where .

We use the following theorem, proof of which is analogous to the proof of Theorem  4.3 in [8], for our error analysis.

Theorem 2.5 (cf. [8], Theorem 4.3). Let be as in (2.13), let be as in (2.14), and let be as in (2.7) with . Then and

3. Discretized Two-Step Newton Method (DTSNM)

We need the following assumptions for the convergence of DTSNM and to obtain the error estimate.

Assumption 3.1 (cf. [7], Assumption 3 (A3)). There exists a constant such that for every and there exists an element such that , .

Assumption 3.2. There exists a continuous, strictly monotonically increasing function with satisfying(i), (ii)(iii) there exists with (cf. [6]) such that(iv) for each there exists a bounded linear operator (cf. [13]) such that with .First we consider a DTSNM for approximating the zero of and then we show that is an approximation to the solution of (1.1) where . For an initial guess and for , the DTSNM is defined as where . Note that with the above notation Let and let be such that

Remark 3.3. Note that the above assumption is satisfied if we choose .Let be the function defined by Let , with

Theorem 3.4. Let and be as in (3.8) and (3.10), respectively, and let and be as in (3.6) and (3.5), respectively, with and . Then the following holds: (a)(b)(c)(d)(e)

Proof. Observe that Now by Assumption 3.1 and (3.7) we have This proves (a). Now (b) follows from (a) and the triangle inequality To prove (c) we observe that where Note that The last but one step follows from Assumption 3.1 and (3.7). Similarly one can prove that Thus from (3.20), (3.22), (3.23), (a) and (b) we have Again since for , for all , by (3.24) we have provided , for all. From (3.26) it is clear that, if . This can be seen as follows: Therefore by (3.27) and (2.9) we have Now since is monotonic increasing and , we have . This completes the proof of the theorem.

Theorem 3.5. Let and the assumptions of Theorem 3.4 hold. Then , for all .

Proof. Note that by (b) of Theorem 3.4 we have that is, . Again note that by (3.29) and (c) of Theorem 3.4 we have that is, . Further by (3.29) and (b) of Theorem 3.4 we have The last but one step follows from the monotonicity of and (3.28).And by (3.31) and (c) of Theorem 3.4 we have that is, . Continuing this way one can prove that , for all . This completes the proof.

The main result of this section is the following theorem.

Theorem 3.6. Let and be as in (3.5) and (3.6), respectively, and let assumptions of Theorem 3.5 hold. Then is a Cauchy sequence in and converges to . Further and where and .

Proof. Using the relation (b) and (e) of Theorem 3.4 and (3.28), we obtain Thus is a Cauchy sequence in and hence it converges, say, to . Observe that from (3.5) Now by letting in (3.35) we obtain . This completes the proof.

Hereafter we assume that and . The proof of the following theorem is analogous to the proof of Theorem  3.14 in [11] but for the sake of completeness we give the proof.

Theorem 3.7 (cf. [11], Theorem 3.14). Suppose is the solution of and Assumptions 3.1 and 3.2 holds, then

Proof. Note that , so Thus where . So by Assumption 3.2, we obtain and hence by (3.39) and (3.40) we have This completes the proof of the theorem.

Theorem 3.8. Suppose is the solution of (3.4) and Assumption 2.1 and Theorem 3.7 hold. In addition if , then

Proof. Suppose and are the solutions of (3.36) and (3.4), respectively, then by (3.36) we have, So from (3.4) and (3.43), Let . Then by (3.44) we have and hence Thus Now the result follows from (2.9), (3.47) and the relation The following theorem is a consequence of Theorems 3.6, 3.7, and 3.8.

Theorem 3.9. Let be as in (3.6) and let assumptions in Theorems 3.6, 3.7 and 3.8 hold. Then where and are as in Theorem 3.6.

Theorem 3.10. Let be as in (3.6) and let assumptions in Theorem 3.9 hold. Further let and Then

4. Algorithm

Note that for ,

Therefore the balancing principle algorithm associated with the choice of the parameter specified in Section 2 involves the following steps:

Step 1. Choose such that and .

Step 2. .

Step 3. Solve for :

Step 4. Solve for ,   .

Step 5. If , then take .

Step 6. Otherwise, repeat with in place of .

Step 7. Choose .

Step 8. Solve using the iteration (3.6).

In the next section we consider an example to illustrate the above algorithm. The computational results provided endorse the reliability and effectiveness of our method.

5. Example

In this section we consider an example satisfying the assumptions made in this paper and give the numerical illustration. Consider the operator where is defined by and defined by where Then for all (see [7], Section ):

Thus the operator is monotone. The Fréchet derivative of is given by

Let be a sequence of finite-dimensional subspaces of and let denote the orthogonal projection on with range . We assume that and as for all . We choose the linear splines in a uniform grid of points in as a basis of .

Since is of the form for some scalars . It can be seen that is a solution of (4.2) if and only if is the unique solution of where

Observe that in Step 4 of algorithm is again in and hence for some . One can see that for is a solution of if and only if is the unique solution of where Compute till and fix . Now choose .

Let ,  ,   and . Then from (3.5) we have where are the grid points.

Observe that is a solution of (3.5) if and only if is the unique solution of where and .

Further from (3.6) it follows that Thus is a solution of (5.14) if and only if is the unique solution of where .

5.1. Numerical Example

Example 5.1. To illustrate the method discussed in the above section, we consider the space and the Fredholm integral operator . The algorithm in Section 5 is applied by choosing as the space of linear splines in a uniform grid of points in . In our computation, we take and . Then the exact solution We use as our initial guess, so that the function satisfies the source condition where . Thus we expect to have an accuracy of order at least .We choose , and approximately. For all the number of iteration in this example. The results of the computation are presented in Table 1. The plots of the exact and the approximate solution obtained are given in Figures 1 and 2.

Table 1

8
4
0.1790
0.0363
0.1388

16
4
0.1729
0.0432
0.1669

32
4
0.1714
0.0450
0.1742

64
4
0.1710
0.0455
0.1761

128
4
0.1709
0.0456
0.1765

256
4
0.1709
0.0456
0.1767

512
4
0.1709
0.0456
0.1767

1024
4
0.1709
0.0456
0.1767

(a)

(b)

(c)

(d)

Figure 1: Curve of the exact and approximate solutions.

(a)

(b)

(c)

(d)

Figure 2: Curve of exact and approximate solutions.

Acknowledgment

M. E. Shobha thanks National Institute of Technology Karnataka India, for the financial support.

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