MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 917972 10.1155/2014/917972 917972 Research Article An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation Cheng Hao Zhu Ping Gao Jie Stenberg Rolf School of Science Jiangnan University Jiangsu Province Wuxi 214122 China jiangnan.edu.cn 2014 2532014 2014 12 01 2014 19 02 2014 25 3 2014 2014 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.

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 , 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 E. However, in general, the a priori bound E cannot be known exactly in practice, and working with a wrong constant E 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: (1)Δu(x,y)+k2u(x,y)=0,x(0,1),y,u(0,y)=φ(y),y,ux(0,y)=0,y, where the constant k>0 is the number of wave. The solution u(x,y) for 0<x<1 will be determined from the noisy data φδ(y). 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 g^(ξ) denote the Fourier transform of the function g(y), which is defined as (2)g^(ξ)=12π-e-iyξg(y)dy,i=-1. The functions φ(y),φδ(y)L2() are the exact and measured data for problem (1), respectively, and satisfy (3)φδ(·)-φ(·)δ, where · denotes the L2-norm and the constant δ>0 is the noise level. Assume that u(x,·)L2() for all 0x<1 and there is the following a priori bound; (4)u(1,·)E, where E is a positive constant.

It is easy to know that for problem (1), (5)u^(x,ξ)=cosh(x|ξ|2-k2)φ^(ξ), and equivalently, (6)u(x,y)=12π-eiyξcosh(x|ξ|2-k2)φ^(ξ)dξ.

Note that the factor cosh(x|ξ|2-k2) increases exponentially for 0<x<1 as |ξ|+; a small distribution for the data φ(x) will be amplified infinitely by this factor and lead to the integral (6) blow-up. Therefore, recovering the temperature u(x,y) from the measured data φδ(x) is severely ill-posed.

For simplicity , we decompose into the following parts I and W, where: (7)I:={ξ,|ξ|k},W:={ξ,|ξ|k}; then L2()=L2(I)L2(W).

For ξW, we can take the regularization approximation solution in the frequency domain as (8)u^mδ(x,ξ)=cos(xk2-|ξ|2)φ^δ(ξ).

For ξI, we introduce an iteration scheme with the following form: (9)u^mδ(x,ξ)=(1-λ)u^m-1δ(x,ξ)+λcosh(x|ξ|2-k2)φ^δ(ξ),hhhhhhhhh            m=1,2,, where λ=λ(ξ)=e-|ξ|2-k2<1 plays an important role in the convergence proof; the initial guess is u^0δ(x,ξ). By using an elementary calculation for (9), we obtain (10)u^mδ(x,ξ)=(1-λ)mu^0δ(x,ξ)+i=0m-1(1-λ)iλcosh(x|ξ|2-k2)φ^δ(ξ)=(1-λ)mu^0δ(x,ξ)+(1-(1-λ)m)  ×cosh(x|ξ|2-k2)φ^δ(ξ). Therefore, the approximate solution of problem (1) has the following form in the frequency domain: (11)u^mδ(x,ξ)={cos(xk2-|ξ|2)φ^δ(ξ),|ξ|k,(1-λ)mu^0δ(x,ξ)+(1-(1-λ)m)×cosh(x|ξ|2-k2)φ^δ(ξ),|ξ|k, or equivalently, (12)umδ(x,y)=12π-eixξu^mδ(x,ξ)dξ, where u^mδ(x,ξ) is given by (11).

Lemma 1 (see [<xref ref-type="bibr" rid="B3">16</xref>]).

For 0<λ1 and m1, the following inequalities hold: (13)(1-λ)mλ1m+1,1-(1-λ)mλm.

Lemma 2.

For 0λ1,0α1 and m1, Lemma 1 can be strengthened as the following inequalities: (14)(1-λ)mλα(m+1)-α,(15)1-(1-λ)mλαmα.

Proof.

In fact, using the established results (13), we can get (16)(1-λ)mλα[(1-λ)mλ]α(m+1)-α,1-(1-λ)mλα[1-(1-λ)mλ]αmα.

Theorem 3.

Let u(x,y) be the exact solution of problem (1) and umδ(x,y) be its regularized approximation given by (12) with u0δ(x,y)=0. Assumptions (3) and (4) are satisfied and one chooses m=E/δ,   where t denotes the largest integer not exceeding t; then there holds the following estimate: (17)u(x,·)-umδ(x,·)3Exδ1-x+δ,0<x<1.

Proof.

Due to the Parseval formula and the triangle inequality, we have (18)u(x,·)-umδ(x,·)2=u^(x,·)-u^mδ(x,·)2=u^(x,·)-u^mδ(x,·)L2(W)2+u^(x,·)-u^mδ(x,·)L2(I)2.

Case  1. While ξW, combining (3), (6), and (12), we have (19)u^(x,·)-u^mδ(x,·)L2(W)=cos(xk2-|ξ|2)φ^(ξ)hhlh-cos(xk2-|ξ|2)φ^δ(ξ)L2(W)φ^(ξ)-φ^δ(ξ)L2(W)φ^(ξ)-φ^δ(ξ)δ.

Case  2. For ξI, combining (4), (6), (12), (14), and (15), we have (20)u^(x,·)-u^mδ(x,·)L2(I)=cosh(x|ξ|2-k2)φ^hhhhh-cosh(x|ξ|2-k2)(1-(1-λ)m)φ^δcosh(x|ξ|2-k2)(1-λ)mφ^+cosh(x|ξ|2-k2)(1-(1-λ)m)(φ^-φ^δ)=cosh(x|ξ|2-k2)(1-λ)mcosh(|ξ|2-k2)cosh(|ξ|2-k2)φ^+cosh(x|ξ|2-k2)(1-(1-λ)m)(φ^δ-φ^)Esupξcosh(x|ξ|2-k2)(1-λ)mcosh(|ξ|2-k2)+δsupξcosh(x|ξ|2-k2)(1-(1-λ)m)2Esupξ(1-λ)me(x-1)|ξ|2-k2+δsupξ(1-(1-λ)m)ex|ξ|2-k2=2Esup0<λ<1λ(1-x)(1-λ)m+δsup0<λ<1λ-x(1-(1-λ)m)2E(m+1)x-1+δmx.

Due to m=E/δ, then mE/δ and m+1E/δ, therefore, (21)u^(x,·)-u^mδ(x,·)L2(I)2E(Eδ)x-1+δ(Eδ)x=3Exδ1-x. 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 0<x<1. The stronger smoothness assumption of u(1,·) may obtain convergence rates for the endpoint x=1; see, for example, , 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: (22)φδ-umδ(0,·)τδφδ-um-1δ(0,·), where τ>1 is a constant and m denotes the regularization parameter. In the numerical experiments, we can take the iteration depth m which satisfies (22) firstly.

Ifu^0δ(x,ξ)=0, thus (22) can be simplified to (23)(1-λ)mφ^δL2(I)τδ(1-λ)m-1φ^δL2(I).

Lemma 5.

The following inequality holds: (24)m2E(τ-1)δ.

Proof.

Due to (4) and (14), we know (25)τδφδ-um-1δ(0,·)=(1-λ)m-1φ^δL2(I)(1-λ)m-1(φ^δ-φ^)L2(I)+(1-λ)m-1φ^L2(I)φ^δ-φ^L2(I)+(1-λ)m-1cosh(|ξ|2-k2)φ^cosh(|ξ|2-k2)L2(I)δ+2Esup0<λ<1(1-λ)m-1λδ+2Em, therefore, (26)m2E(τ-1)δ.

Lemma 6.

Setting ωm(x,·)=u(x,·)-um(x,·), then the following inequality holds: (27)ωm(x,·)L2(I)2ωm(1,·)L2(I)xωm(0,·)L2(I)1-x.

Proof.

Defining um(x,·)=(1-(1-λ)m)cosh(x|ξ|2-k2)φ^(ξ), then we have (28)ωm(0,·)=(1-λ)mφ^,ωm(1,·)=(1-λ)mcosh(|ξ|2-k2)φ^.ωm(x,·)L2(I)2=(1-λ)mcosh(x|ξ|2-k2)φ^L2(I)2=ξI(1-e-|ξ|2-k2)2m(cosh(x|ξ|2-k2)φ)2dξξI(1-e-|ξ|2-k2)2m(ex|ξ|2-k2φ)2dξ=ξI(1-e|ξ|2-k2)2mx(e|ξ|2-k2φ)2xHH×(1-e|ξ|2-k2)2m(1-x)(e|ξ|2-k2φ)2(1-x)dξ(ξI((1-e-|ξ|2-k2)me|ξ|2-k2φ)2dξ)x×(ξI((1-e-|ξ|2-k2)mφ)2dξ)1-x4(ξI((1-e-|ξ|2-k2)mcosh(|ξ|2-k2)φ)2dξ)x×(ξI((1-e-|ξ|2-k2)mφ)2dξ)1-x=4ωm(1,·)L2(I)2xωm(0,·)L2(I)2(1-x).

Lemma 7.

The following inequality holds: (29)um(x,·)-u(x,·)L2(I)2(τ+1)1-xExδ1-x.

Proof.

Due to (3), (4), and (27), we know that (30)um(1,·)-u(1,·)L2(I)=(1-λ)mcosh(|ξ|2-k2)φE,um(0,·)-u(0,·)L2(I)=(1-λ)mφL2(I)(1-e-|ξ|2-k2)m(φ-φδ)L2(I)+(1-e-|ξ|2-k2)mφδL2(I)(τ+1)δ. Combining (30), we have (31)um(x,·)-u(x,·)L2(I)2(τ+1)1-xExδ1-x.

Theorem 8.

Let u(x,y) be the exact solution of problem (1) and umδ(x,y) be its regularization approximation defined by (12) with u0δ(x,y)=0. If the a priori bound (4) is valid and the iteration (9) is stopped by the discrepancy principle (22), then (32)umδ(x,·)-u(x,·)CEy/y0δ1-(y/y0)+δ, where C=(2/(τ-1))x+2(τ+1)1-x.

Proof.

According to the triangle inequality, (19), (24), and (29), we obtain that (33)umδ(x,·)-u(x,·)umδ(x,·)-u(x,·)L2(I)+umδ(x,·)-u(x,·)L2(W)umδ(x,·)-um(x,·)L2(I)+um(x,·)-u(x,·)L2(I)+δδmx+2(τ+1)1-xExδ1-x+δδ(2E(τ-1)δ)x+2(τ+1)1-xExδ1-x+δ=((2τ-1)x+2(τ+1)1-x)Exδ1-x+δ.

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 ayb,  N denotes the number of discrete points.

For an exact data function φ(y), its discrete noisy version is (34)φϵ=φ+ϵrandn(size(φ)), where (35)φ=(φ(y1),,φ(yN)),yj=a+(b-a)(j-1)N+1,j=1,2,N,δ=φϵ-φl2:=1Ni=1N|φϵ(yi)-φ(yi)|2. The function “randn(·)” generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1. The absolute error ea(u) and the relative error er(u) are defined by (36)ea(u):=umδ(x,·)-u(x,·)l2,(37)er(u):=umδ(x,·)-u(x,·)l2u(x,·)l2, respectively.

In the numerical experiment, we compute the approximation umδ(x,y) according to Theorem 3. And we can take the discrete points N=100, the number of wave k=1, a priori bound E=u(1,·)l2()0.7, and a priori parameter m=E/δ. The a posteriori parameter m was chosen according to formula (22) and τ=2 for calculation. Meanwhile, we take a=-6 and b=6 in the first numerical example. For Example 2, we take a=-4 and b=4.

Example 1.

If we take the function φ(y)=e-y2𝒮(), where 𝒮() denotes the Schwartz function space, φ^(ξ)𝒮() decays rapidly and formula (6) can be used to calculate u(x,y) with exact data directly. To observe the effect on different noise levels ϵ, we only take the case of k=1 at x=0.9.

Table 1 shows the comparison of the errors between the exact and regularization solutions for different ϵ, from which we can see that the smaller the ϵ is, the better the computed approximation is.

The errors between the exact and approximate solutions of Example 1, with k=1 at x=0.9.

ϵ 1 e - 4 1 e - 3 1 e - 2 1 e - 1
m 4909 534 51 4
e a ( u ) 0.4435 0.5194 0.8220 2.1846
e r ( u ) 0.1021 0.1196 0.1893 0.5031

Figure 1 is the comparison of a priori and a posteriori parameter choice rules for the exact u(x,y) and the approximate solution ukδ(x,y) at x=0.1,0.3,0.6,0.9 for the noise level ϵ=10-2. Here we also take the reasonable a priori bound E=u(1,·)l2(), and we can see that the a posteriori rule also works effectively.

Example 1. The regularization solution with a priori and a posteriori parameter choice rules for the noise level ϵ=10-2. (a) y=0.1, (b) y=0.3, (c) y=0.6, and (d) y=0.9, respectively.

Example 2.

The function (38)u(x,y)=1π2sin(πky)cosh(kπ2-1x) is the exact solution of problem (1) with the Cauchy data u(0,y)=φ(y)=(1/π2)sin(πky) and ux(0,y)=0.

Figure 2 is the comparison of the exact solution u(x,y) and the approximation umδ(x,y) at different points x=0.1,0.3,0.6,0.9 and noise levels ϵ=10-2,10-3 for the a priori parameter choice rule. Here we take the a priori bound E=u(1,·)l2(), and the proposed method works well for the a priori parameter choice rule.

Example 2. The exact and regularized solutions for the different noise levels. (a) x=0.1, (b) x=0.3, (c) x=0.6, and (d) x=0.9  ϵ=10-2,10-3, respectively.

Figure 3 is the comparison of the different a priori bound E=0.07,0.7,7 for the different points x=0.1,0.3,0.6,0.9 at the noise level ϵ=10-2. From this figure we can see that working with a wrong constant E would lead to a bad regularized solution. Therefore, a reasonable a priori bound E is very important for the a priori parameter choice rule.

Example 2. The exact and regularized solutions at (a) x=0.1, (b) x=0.3, (c) x=0.6, and (d) x=0.9 for the same noise level ϵ=10-2 but different a priori bounds E=0.07,0.7,7, respectively.

Figure 4 is the comparison of a priori and a posteriori parameter choice rules for the exact u(x,y) and the approximate solution umδ(x,y) at x=0.1,0.3,0.6,0.9 for the noise level ϵ=10-2. Here we take the reasonable a priori bound E=u(1,·)l2() as previously mentioned, and the a posteriori rule also works effectively as expected.

Example 2. The regularization solution with a priori and a posteriori parameter choice rules for the noise level ϵ=10-2. (a) x=0.1, (b) x=0.3, (c) x=0.6, and (d) x=0.9, respectively.

From Figures 14, we concluded that the smaller the ϵ is, the better the computed approximation is, and the bigger the x is, the worse the computed approximation is. In addition, the a priori bound E has great influence on the numerical results. Although the a posteriori regularization parameter selection rule does not rely on a priori bound E, 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).

Beskos D. E. Boundary element methods in dynamic analysis: part II (1986–1996) Applied Mechanics Reviews 1997 50 3 149 197 2-s2.0-0031095811 Chen J. T. Wong F. C. Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition Journal of Sound and Vibration 1998 217 1 75 95 2-s2.0-0032180927 Harari I. Barbone P. E. Slavutin M. Shalom R. Boundary infinite elements for the Helmholtz equation in exterior domains International Journal for Numerical Methods in Engineering 1998 41 6 1105 1131 2-s2.0-0032024388 Hall W. S. Mao X. Q. A boundary element investigation of irregular frequencies in electromagnetic scattering Engineering Analysis with Boundary Elements 1995 16 3 245 252 2-s2.0-0029387052 Marin L. Elliott L. Heggs P. J. Ingham D. B. Lesnic D. Wen X. An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation Computer Methods in Applied Mechanics and Engineering 2003 192 5-6 709 722 2-s2.0-0037474371 10.1016/S0045-7825(02)00592-3 MR1952356 ZBL1022.78012 Marin L. Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations Computers and Structures 2005 83 4-5 267 278 2-s2.0-11344277052 10.1016/j.compstruc.2004.10.005 MR2115987 Marin L. Elliott L. Heggs P. J. Ingham D. B. Lesnic D. Wen X. Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation International Journal for Numerical Methods in Engineering 2004 60 11 1933 1947 2-s2.0-3242796795 10.1002/nme.1031 MR2070433 ZBL1062.78015 Marin L. Elliott L. Heggs P. J. Ingham D. B. Lesnic D. Wen X. BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method Engineering Analysis with Boundary Elements 2004 28 9 1025 1034 2-s2.0-3142686215 10.1016/j.enganabound.2004.03.001 Regińska T. Wakulicz A. Wavelet moment method for the Cauchy problem for the Helmholtz equation Journal of Computational and Applied Mathematics 2009 223 1 218 229 2-s2.0-54249096190 10.1016/j.cam.2008.01.005 MR2463112 ZBL1165.65069 Qin H.-H. Wei T. Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation Mathematics and Computers in Simulation 2009 80 2 352 366 2-s2.0-70349766700 10.1016/j.matcom.2009.07.005 MR2582118 ZBL1185.65172 Feng X. L. Fu C. L. Cheng H. A modified Tikhonov regularization for solving the Cauchy problem for the Helmholtz equation http://www.mai.liu.se/xifen Qin H. H. Wei T. Shi R. Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation Journal of Computational and Applied Mathematics 2009 224 1 39 53 2-s2.0-56449120046 10.1016/j.cam.2008.04.012 MR2474210 ZBL1158.65072 Fu C.-L. Feng X.-L. Qian Z. The Fourier regularization for solving the Cauchy problem for the Helmholtz equation Applied Numerical Mathematics 2009 59 10 2625 2640 2-s2.0-67649831323 10.1016/j.apnum.2009.05.014 MR2553158 ZBL1169.65333 Regińska T. Regiński K. Approximate solution of a Cauchy problem for the Helmholtz equation Inverse Problems 2006 22 3 975 989 2-s2.0-33744543954 10.1088/0266-5611/22/3/015 MR2235649 ZBL1099.35160 Regińska T. Tautenhahn U. Conditional stability estimates and regularization with applications to cauchy problems for the helmholtz equation Numerical Functional Analysis and Optimization 2009 30 9-10 1065 1097 2-s2.0-73449116096 10.1080/01630560903393170 MR2589765 ZBL1181.47009 Cheng H. Fu C.-L. An iteration regularization for a time-fractional inverse diffusion problem Applied Mathematical Modelling 2012 36 11 5642 5649 10.1016/j.apm.2012.01.016 MR2956772 ZBL1254.65100 Engl H. W. Hanke M. Neubauer A. Regularization of Inverse Problems 1996 Boston, Mass, USA Kluwer Academic 10.1007/978-94-009-1740-8 MR1408680 Nair M. T. Schock E. Tautenhahn U. Morozov's discrepancy principle under general source conditions Zeitschrift für Analysis und ihre Anwendungen 2003 22 1 199 214 2-s2.0-0037227075 MR1962084 ZBL1037.65057