Research Article | Open Access
An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation
A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. The a priori and a posteriori rules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.
The Cauchy problem for the Helmholtz equation arises naturally in many areas of engineering and science, especially in wave propagation and vibration phenomena, such as the vibration of a structure , the acoustic cavity problem , the radiation wave , and the scattering of a wave . However, this problem is severely ill-posed in the sense that a small change in the Cauchy data would lead to a dramatic variation in the solution. Therefore, it is necessary to study different highly efficient algorithms to solve this problem. Recently, a few special numerical methods to deal with this problem have been developed, such as the boundary element method , the method of fundamental solutions , the conjugate gradient method , the Landweber method , wavelet moment method , quasi-reversibility and truncation methods , modified Tikhonov regularization method [11, 12], the fourier regularization method , and so forth [9, 14, 15]. However, most of them choose the regularization parameter by the a priori rule, which depends seriously on the a priori bound . However, in general, the a priori bound cannot be known exactly in practice, and working with a wrong constant may lead to a bad regularized solution. Therefore, giving the a posteriori parameter choice rule is a very meaningful topic.
In this paper we will consider the following problem with inhomogeneous Dirichlet data in a strip domain: where the constant is the number of wave. The solution for will be determined from the noisy data . In this paper a regularization method of iteration type for solving this problem will be given. By dint of this method, the a priori and a posteriori rule for choosing a regularization parameter with strict theory analysis, as well as order optimal error estimates, will be obtained.
The outline of the paper is as follows. In Section 2, an order optimal error estimate is obtained for the a priori parameter choice rule. The a posteriori parameter choice rule is given in Section 3, which also leads to a Hölder-type error estimate. Numerical implement shows the effectiveness of the proposed method in Section 4.
2. Regularization and Error Estimate
Let denote the Fourier transform of the function , which is defined as The functions are the exact and measured data for problem (1), respectively, and satisfy where denotes the -norm and the constant is the noise level. Assume that for all and there is the following a priori bound; where is a positive constant.
It is easy to know that for problem (1), and equivalently,
Note that the factor increases exponentially for as ; a small distribution for the data will be amplified infinitely by this factor and lead to the integral (6) blow-up. Therefore, recovering the temperature from the measured data is severely ill-posed.
For simplicity , we decompose into the following parts and , where: then .
For , we can take the regularization approximation solution in the frequency domain as
For , we introduce an iteration scheme with the following form: where plays an important role in the convergence proof; the initial guess is . By using an elementary calculation for (9), we obtain Therefore, the approximate solution of problem (1) has the following form in the frequency domain: or equivalently, where is given by (11).
Lemma 1 (see ). For and , the following inequalities hold:
Lemma 2. For and , Lemma 1 can be strengthened as the following inequalities:
Proof. In fact, using the established results (13), we can get
Theorem 3. Let be the exact solution of problem (1) and be its regularized approximation given by (12) with . Assumptions (3) and (4) are satisfied and one chooses , where denotes the largest integer not exceeding ; then there holds the following estimate:
Proof. Due to the Parseval formula and the triangle inequality, we have
Case 1. While , combining (3), (6), and (12), we have
Case 2. For , combining (4), (6), (12), (14), and (15), we have
Due to , then and , therefore, Combing inequalities (18), (19), and (21), the proof of this theorem is completed.
Remark 4. Obviously, Theorem 3 could only solve the problem with the case . The stronger smoothness assumption of may obtain convergence rates for the endpoint ; see, for example, [10–12], and we omit the further discussions.
3. The Discrepancy Principle
In this section, we discuss an a posteriori stopping rule for iterative scheme (9) which is based on the discrepancy principle of Morozov [17, 18] in the following form: where is a constant and denotes the regularization parameter. In the numerical experiments, we can take the iteration depth which satisfies (22) firstly.
If, thus (22) can be simplified to
Lemma 5. The following inequality holds:
Lemma 6. Setting , then the following inequality holds:
Proof. Defining , then we have
Lemma 7. The following inequality holds:
Theorem 8. Let be the exact solution of problem (1) and be its regularization approximation defined by (12) with . If the a priori bound (4) is valid and the iteration (9) is stopped by the discrepancy principle (22), then where .
4. Numerical Test
In this section, a simple numerical example is devised to verify the validity of the proposed method. We use the discrete Fourier transform and inverse Fourier transform (or FFT and IFFT algorithms) to complete our numerical experiment. We fix the interval , denotes the number of discrete points.
For an exact data function , its discrete noisy version is where The function “” generates arrays of random numbers whose elements are normally distributed with mean 0, variance . The absolute error and the relative error are defined by respectively.
In the numerical experiment, we compute the approximation according to Theorem 3. And we can take the discrete points , the number of wave , a priori bound , and a priori parameter . The a posteriori parameter was chosen according to formula (22) and for calculation. Meanwhile, we take and in the first numerical example. For Example 2, we take and .
Example 1. If we take the function , where denotes the Schwartz function space, decays rapidly and formula (6) can be used to calculate with exact data directly. To observe the effect on different noise levels , we only take the case of at .
Table 1 shows the comparison of the errors between the exact and regularization solutions for different , from which we can see that the smaller the is, the better the computed approximation is.
Figure 1 is the comparison of a priori and a posteriori parameter choice rules for the exact and the approximate solution at for the noise level . Here we also take the reasonable a priori bound , and we can see that the a posteriori rule also works effectively.
Example 2. The function is the exact solution of problem (1) with the Cauchy data and .
Figure 2 is the comparison of the exact solution and the approximation at different points and noise levels for the a priori parameter choice rule. Here we take the a priori bound , and the proposed method works well for the a priori parameter choice rule.
Figure 3 is the comparison of the different a priori bound for the different points at the noise level . From this figure we can see that working with a wrong constant would lead to a bad regularized solution. Therefore, a reasonable a priori bound is very important for the a priori parameter choice rule.
Figure 4 is the comparison of a priori and a posteriori parameter choice rules for the exact and the approximate solution at for the noise level . Here we take the reasonable a priori bound as previously mentioned, and the a posteriori rule also works effectively as expected.
From Figures 1–4, we concluded that the smaller the is, the better the computed approximation is, and the bigger the is, the worse the computed approximation is. In addition, the a priori bound has great influence on the numerical results. Although the a posteriori regularization parameter selection rule does not rely on a priori bound , it also works well.
In this paper an iteration regularization method is given for solving the numerical analytic continuation problem on a strip domain. The a priori and a posteriori rules for choosing a regularization parameter with strict theory analysis are presented. In numerical aspect, the comparison with different parameter choice rules shows that the proposed method works effectively.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is supported by the Natural Science Foundation of Jiangsu Province of China for Young Scholar (no. BK20130118), the Fundamental Research Funds for the Central Universities (no. JUSRP1033), and the NNSF of China (nos. 11171136, 11271163, and 11371174).
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