Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 953492, 14 pagesdoi:10.1155/2011/953492

Research Article

The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

Fan Yang,¹ HengZhen Guo,² and XiaoXiao Li¹

¹School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China²Institute of Education, Lanzhou City University, Lanzhou, Gansu 730070, China

Received 20 December 2010; Revised 6 April 2011; Accepted 25 April 2011

Academic Editor: Paulo Batista Gonçalves

Copyright © 2011 Fan Yang et al. 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.

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

1. Introduction

Inverse source problems arise in many branches of science and engineering, for example, heat conduction, crack identification, electromagnetic theory, geophysical prospecting, and pollutant detection. For the heat source identification, there has been a large number of research results for different forms of heat source [1–6]. To the author's knowledge, there were few papers for identifying the unknown source on the modified Helmholtz equation which is pointed out in [7] by regularization method.

In this paper, we consider the following inverse problem: to find a pair of functions satisfying where is the unknown source depending only on one spatial variable, is the supplementary condition, and the constant is the wave number. In applications, input data can only be measured, and there will be measured data function which is merely in and satisfies where the constant represents a noise level of input data.

It is easy to derive a solution of problem (1.1) by the method of separation of variables where is an orthogonal basis in , and By the supplementary condition, we define the operator , then we have It is easy to see that is a linear compact operator and the singular values of satisfy that is, where Therefore,

Note that as , thus the exact data function must satisfy the property that decays rapidly as . As for the measured data function is only in , we cannot expect the coefficient of has the same decay rate. Thus, the problem (1.1) is ill posed. It is impossible to gain the unknown source using classical methods. In the following sections, we will use the simplified Tikhonov method to deal with the ill posed problem. Before doing that, we impose an a priori bound on the unknown source; that is, where is a constant and denotes the norm in Sobolev space which is defined by [8] as follows:

The simplified Tikhonov regularization method was based on the Tikhonov regularization method. Skillfully simplifying the filter gained by the Tikhonov regularization, a better regularization approximation solution of the inverse problem was obtained. This idea initially came from Carasso, the author who modified the filter gained by the Tikhonov regularization method and obtained the order optimal error estimate in [9]. By this method, Fu [10] considered the inverse heat conduction problem on a general sideways parabolic equation, and Cheng et al. [11, 12] considered the spherically symmetric inverse problem.

This paper is organized as follows. Section 2 gives some auxiliary results. Section 3 gives a simplified Tikhonov regularization solution and error estimation. Section 4 gives two examples to illustrate the accuracy and efficiency of this methods. Section 5 puts an end to this paper with a brief conclusion.

2. Some Auxiliary Results

Now, we give some important Lemmas, which are very useful for our main conclusion.

Lemma 2.1. For and is a positive constant, there holds

Lemma 2.2. For , there holds the following inequalities:

Proof. Let The proof of (2.2) can be separated from two cases.Case 1. For large values of , that is, , we get Case 2. , we obtain If , above inequality becomes into If , we get Combining (2.4) with (2.6) and (2.7), the first inequality equation is obtained.Let Using Lemma 1, we obtain Let then Setting , we can obtain . It is easy to see that is a unique maximal value point of . So, So, we get This completes the proof.

3. A Simplified Tikhonov Regularization Method

Since problem (1.1) is an ill-posed problem, we give an approximate solution of by a Tikhonov regularization method which minimizes the quantity Then, by Theorem 2.12 in [8], the unique solution of the minimization problem (3.1) is equal to solve the following normal equation: that is, Because is a linear self-adjoint compact operator, that is, , we have the equivalent form We define function of a compact self-adjoint operator by the spectral mapping theorem in the following way.

Definition 3.1 (see[13]). If is a real-valued continuous function on the spectrum , we define by where is a compact self-adjoint, , and are the corresponding orthogonal eigenvectors.

So, we obtain

Comparing (1.10) with (3.6), we can find that the procedure consists in replacing the unknown with an appropriately filtered noised data . The filter in (3.6) attenuates the coefficient of in a manner consistent with the goal of minimizing quantity (3.1). By this idea, we can use a much better filter to replace the filter and give another approximation of the solution .

We define a regularization approximate solution of problem (1.1) for noisy data which is called the simplified Tikhonov regularized solution of problem (1.1) as follows:

Theorem 3.2. Let be the simplified Tikhonov approximation of the solution of problem (1.1). Let be measured data at satisfying (1.2), and let priori condition (1.11) hold for . If one selects then the following estimate holds:

Proof. Due to the triangle inequality, we have The proof is complete.

Remark 3.3. If , If , Hence, can be viewed as the approximation of the exact solution .

4. Numerical Example

From (1.10), we know that

We use trapezoid's rule to approach the integral and do an approximate truncation for the series by choosing the sum of the front terms. After considering an equidistant grid ), we get where

Example 4.1. It is easy to see that the function and the function are the exact solutions of the problem (1.1). Consequently, the data function is , and Adding a random distributed perturbation to each data function, we obtain vector ; that is, The function “” generates arrays of random numbers whose elements are normally distributed with mean 0, variance , and standard deviation . “Randn(size(g))” returns an array of random entries that is the same size as . The total noise level can be measured in the sense of root mean square error (RMSE) according to

Using as data function, we obtain the computed approximation by using (3.7). The relative error is given as follows: where is defined by (4.6).

Tables 1-2 show and have small influence on the relative error when they become larger. So, we will always take and in the following examination.

Table 1: Relative errors with , , and for different .

10
50
100
200
400
800
1600
3200

0.0199
0.0080
0.0041
0.0046
0.0045
0.0033
0.0013
0.0010

Table 2: Relative errors with , , and for different .

1
3
4
6
7
8
10
11

1
0.0019
0.0022
0.0026
0.0035
0.0044
0.0034
0.0055

Test 1We choose , , and in Tables 3, 4, 5, and 6 to compute the parameters , , and , respectively.

Table 3: , and with , and .

0.0332
0.0331

0.0510
0.0231
0.0109
0.0049
0.0022

0.1018
0.0529
0.0607
0.0621
0.0624

Table 4: , and with , and .

0.1
0.05
0.01
0.005
0.001

0.0287
0.0150
0.0029
0.0015

0.0067
0.0040
0.0011

0.0376
0.0156
0.0035
0.0015

Table 5: , and with , and .

0.0300
0.0151
0.0028
0.0014

0.0245
0.0174
0.0075
0.0052
0.0024

0.0300
0.0105
0.0036

Table 6: , , and with , and .

0.0293
0.0133
0.0028
0.0015

0.0369
0.0269
0.0144
0.0111
0.0058

0.0352
0.0135
0.0051
0.0024

These tables indicate that parameter , , and all depend on the perturbation . , , and decrease with the decrease of . These are consistent with our regularization methods. In addition, decreases with the increase of at first, but it ceases decreasing when reaches to some extent. This means that does not decrease for stronger “smoothness” assumptions on the exact solution .

Test 2Figure 1 shows the comparison between the exact solution and the regularization solution for , , , and with the perturbation , , and with Example 4.1.

(a)

(b)

(c)

(d)

Figure 1: The exact solution and its approximation : (a) , (b) , (c) , and (d) .

Figure 1 indicates these regularized solutions approximate to the exact solution, as the amount of decreases, while the numerical results are not so good as the parameter becomes larger.

Test 3The unknown sources is given. The numerical test was constructed in the following way. First, we selected the solution and obtained the exact data function using (1.6). Then, we added a normally distributed perturbation to each data function giving vectors . Finally, we obtained the regularization solutions using (3.7).

Example 4.2. Consider a piecewise smooth source:

In Example 4.2, since the direct problem with the heat source does not have an analytical solution, the data is obtained by solving the direct problem. Figures 2-3 show the comparisons between the exact solution and its computed approximation with different noise level for Example 4.2. It can be seen that as the amount of noise decreases, the regularized solutions approximate better the exact solution. In addition, even when , the regularized solutions still approximate the exact solution. From Figures 2-3, it can be seen that the numerical solution is less than that of Example 4.1. It is not difficult to see that the well-known Gibbs phenomenon and the recovered data near the nonsmooth points are not accurate. Taking into consideration the ill posedness of the problem, the results presented in Figures 2-3 are reasonable.

(a)

(b)

Figure 2: The exact solution and its approximation for : (a) , and (b) .

(a)

(b)

Figure 3: The exact solution and its approximation for : (a) , and (b) .

5. Conclusions

In this paper, we considered the inverse problem of determining the unknown source using the simplified Tikhonov regularization method for the modified Helmholtz equation. It was shown that with a certain choice of the parameter, a stability estimate was obtained. Meanwhile, the numerical example verified the efficiency and accuracy of this method.

Acknowledgment

The project is supported by the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015).

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