Mathematical Problems in Engineering

Volume 2014, Article ID 917972, 9 pages

http://dx.doi.org/10.1155/2014/917972

## An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

School of Science, Jiangnan University, Jiangsu Province, Wuxi 214122, China

Received 12 January 2014; Accepted 19 February 2014; Published 25 March 2014

Academic Editor: Rolf Stenberg

Copyright © 2014 Hao Cheng 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

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.

#### 1. Introduction

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 [1], the acoustic cavity problem [2], the radiation wave [3], and the scattering of a wave [4]. 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 [5], the method of fundamental solutions [6], the conjugate gradient method [7], the Landweber method [8], wavelet moment method [9], quasi-reversibility and truncation methods [10], modified Tikhonov regularization method [11, 12], the fourier regularization method [13], 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 [15], 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 [16]). *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:
*

*Proof. *Due to (4) and (14), we know
therefore,

Lemma 6. *Setting , then the following inequality holds:
*

*Proof. *Defining , then we have

Lemma 7. *The following inequality holds:
*

*Proof. *Due to (3), (4), and (27), we know that
Combining (30), we have

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 .*

*Proof. *According to the triangle inequality, (19), (24), and (29), we obtain that

#### 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.

#### 5. Conclusion

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.

#### Acknowledgment

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).

#### References

- D. E. Beskos, “Boundary element methods in dynamic analysis: part II (1986–1996),”
*Applied Mechanics Reviews*, vol. 50, no. 3, pp. 149–197, 1997. View at Google Scholar · View at Scopus - J. T. Chen and F. C. Wong, “Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition,”
*Journal of Sound and Vibration*, vol. 217, no. 1, pp. 75–95, 1998. View at Google Scholar · View at Scopus - I. Harari, P. E. Barbone, M. Slavutin, and R. Shalom, “Boundary infinite elements for the Helmholtz equation in exterior domains,”
*International Journal for Numerical Methods in Engineering*, vol. 41, no. 6, pp. 1105–1131, 1998. View at Google Scholar · View at Scopus - W. S. Hall and X. Q. Mao, “A boundary element investigation of irregular frequencies in electromagnetic scattering,”
*Engineering Analysis with Boundary Elements*, vol. 16, no. 3, pp. 245–252, 1995. View at Google Scholar · View at Scopus - L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 192, no. 5-6, pp. 709–722, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Marin and D. Lesnic, “The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations,”
*Computers and Structures*, vol. 83, no. 4-5, pp. 267–278, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation,”
*International Journal for Numerical Methods in Engineering*, vol. 60, no. 11, pp. 1933–1947, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, “BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method,”
*Engineering Analysis with Boundary Elements*, vol. 28, no. 9, pp. 1025–1034, 2004. View at Publisher · View at Google Scholar · View at Scopus - T. Regińska and A. Wakulicz, “Wavelet moment method for the Cauchy problem for the Helmholtz equation,”
*Journal of Computational and Applied Mathematics*, vol. 223, no. 1, pp. 218–229, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H.-H. Qin and T. Wei, “Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation,”
*Mathematics and Computers in Simulation*, vol. 80, no. 2, pp. 352–366, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. L. Feng, C. L. Fu, and H. Cheng, “A modified Tikhonov regularization for solving the Cauchy problem for the Helmholtz equation,” http://www.mai.liu.se/xifen.
- H. H. Qin, T. Wei, and R. Shi, “Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation,”
*Journal of Computational and Applied Mathematics*, vol. 224, no. 1, pp. 39–53, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - C.-L. Fu, X.-L. Feng, and Z. Qian, “The Fourier regularization for solving the Cauchy problem for the Helmholtz equation,”
*Applied Numerical Mathematics*, vol. 59, no. 10, pp. 2625–2640, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Regińska and K. Regiński, “Approximate solution of a Cauchy problem for the Helmholtz equation,”
*Inverse Problems*, vol. 22, no. 3, pp. 975–989, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Regińska and U. Tautenhahn, “Conditional stability estimates and regularization with applications to cauchy problems for the helmholtz equation,”
*Numerical Functional Analysis and Optimization*, vol. 30, no. 9-10, pp. 1065–1097, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Cheng and C.-L. Fu, “An iteration regularization for a time-fractional inverse diffusion problem,”
*Applied Mathematical Modelling*, vol. 36, no. 11, pp. 5642–5649, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. W. Engl, M. Hanke, and A. Neubauer,
*Regularization of Inverse Problems*, Kluwer Academic, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - M. T. Nair, E. Schock, and U. Tautenhahn, “Morozov's discrepancy principle under general source conditions,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 22, no. 1, pp. 199–214, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus