MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 878109 10.1155/2012/878109 878109 Research Article A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation Gao Jinghuai 1 Wang Dehua 1 Peng Jigen 2 Roque Carla 1 Institute of Wave and Information National Engineering Laboratory for Offshore Oil Exploration School of Electronic and Information Engineering Xi'an Jiaotong University, Xi'an, Shaanxi 710049 China xjtu.edu.cn 2 School of Mathematics and Statistics Xi'an Jiaotong University Xi'an, Shaanxi 710049 China xjtu.edu.cn 2012 24 12 2012 2012 30 09 2012 16 11 2012 22 11 2012 2012 Copyright © 2012 Jinghuai Gao 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.

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed method.

1. Introduction

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. It has a wide range of applications, for example, radar, sonar, geographical exploration, and medical imaging. A kind of important equations similar with the Helmholtz equation in science and engineering is (1.1)Δu(x,y)-k2u(x,y)=f(x), where the constant k>0 is the wave number and f(x) is the source term. This equation is called the modified Helmholtz equation. It appears, for example, in the semi-implicit temporal discretization of the heat or the Navier-Stokes equations  and in the linearized Poisson-Boltzmann equation.

Inverse source problems have attracted great attention of many researchers over recent years because of their applications to many practical problems such as crack determination [2, 3], heat source determination , inverse heat conduction , pollution source identification , electromagnetic source identification , Stefan design problems , sound source reconstruction , and identification of current dipolar sources in the so-called inverse electroencephalography/magnetoencephalography (EEG/MEG) problems [16, 17]. Theoretical investigation on the inverse source identification problems can be found in the works of .

The main difficulty of inverse source identification problems is that they are typically ill posed in the sense of Hadamard . In other words, any small error in the scattered measurement data may induce enormous error to the solution. In general, the unknown source can only be recovered from boundary measurements if some a priori knowledge is assumed. For instances, if one of the products in the separation of variables is known [25, 26], the base area of a cylindrical source is known , or a nonseparable type is in the form of a moving front , then the boundary condition plus some scattered boundary measurements can uniquely determine the unknown source term. Furthermore, when the unknown source term is relatively smooth, some regularization techniques can be employed, see [5, 2729] for more details. In addition, due to the complexity and ill posedness of the inverse source identification problems, some of the variational methods [6, 30] are also employed to deal with them.

In this paper, we will consider the following problem (see ): (1.2)Δu(x,y)-k2u(x,y)=f(x),0<x<π,0<y<,u(0,y)=u(π,y)=0,0y<,u(x,0)=0,0xπ,u(x,y)|y  bounded,0xπ,u(x,1)=g(x),0xπ, for determining the source term f(x) such that the solution u(x,y) of the modified Helmholtz equation satisfies the given supplementary condition u(x,1)=g(x), where the constant k>0 is the wave number. In practice, the data g(x) is usually obtained through measurement and the measured data is denoted by gδ(x).

To determine the source term f(x), we require the following assumptions:

(A) f(x)L2(0,π) and g(x)L2(0,π);

(B) there exists a relation between the function g(x) and the measured data gδ(x): (1.3)g-gδL2δ,

where ·L2 denotes the norm in the space L2(0,π) and δ>0 is the noise level.

(C) The source term f(x) satisfies the a priori bound (1.4)fHpE,      p0,

where E is a positive constant, and ·Hp denotes the norm in Sobolev space Hp(0,π) which is defined by  as follows: (1.5)f(·)Hp=(n=1(1+n2)p|f,Xn|2)1/2.

We can refer to  for the details of the Sobolev space Hp(0,π).

Using the separation of variables, we can obtain the explicit solution of the modified Helmholtz equation: (1.6)u(x,y)=-n=11-e-n2+k2yn2+k2fnXn(x), where (1.7){Xn(x)=2πsin(nx),n=1,2,} is an orthogonal basis in L2(0,π), and (1.8)fn=2π0πf(x)sin(nx)dx. According to (1.6) and the supplementary condition u(x,1)=g(x), we have (1.9)g(x)=-n=11-e-n2+k2n2+k2fnXn(x). Based on this relation, we can define an operator K on the space L2(0,π) as (1.10)(Kf)(x)=-n=11-e-n2+k2n2+k2fnXn(x). Thus, our inverse source problem is formulated as follows: give the data g(x) and the operator K and then determine the unknown source term f(x) such that (1.10) holds.

It is straightforward that the operator K is invertible and then the exact solution of (1.2) is (1.11)f(x)=(K-1g)(x)=-n=1n2+k21-e-n2+k2gnXn(x), with (1.12)gn=2π0πg(x)sin(nx)dx. Note that the factor (n2+k2)/(1-e-n2+k2) increases rapidly and tends to infinity as n, so a small perturbation in the data g(x) may cause a dramatically large error in the solution f(x). Therefore, this inverse source problem is mildly ill posed. It is impossible to obtain the unknown source using classical methods as above.

In , the simplified Tikhonov regularization method was given for (1.2). In this method, the regularization parameter is a priori chosen. It is well known that the ill-posed problem is usually sensitive to the regularization parameter and the a priori bound is usually difficult to be obtained precisely in practice. So the a priori choice rule of the regularization parameter is unreliable in practical problems. In this paper, we will present a Tikhonov-type regularization method to deal with (1.2) and show that the regularization parameter can be chosen by an a posteriori rule based on the discrepancy principle in .

The rest of this paper is organized as follows. In Section 2, we establish a quasinormal equation, which is crucial for proving the convergence of the Tikhonov-type regularization method. In Section 3, we give the Tikhonov-type regularization method and then prove the convergence of such method. Also, we give a priori and a posteriori choice rules to find the regularization parameter in the regularization method. In Section 4, we demonstrate a numerical example to illustrate the effectiveness of the method. In Section 5, we give some conclusions.

2. Preparation

In this section, we give an auxiliary result which will be used in this paper.

We first define an operator T on L2(0,π) as follows: (2.1)(Tf)(x)=n=1(1+n2)pfnXn(x). Let us observe that the operator T is well defined and is a self-adjoint linear operator.

Next we give a lemma, which is important for discussing the regularization method. For simplicity, we denote the spaces Hp(0,π) and L2(0,π) by X and Y, respectively.

Lemma 2.1.

Let A:XY be a linear and bounded operator between two Hilbert spaces, and let T be defined as in (2.1), α>0. Then for any xX the following Tikhonov functional (2.2)Jα(x):=Ax-yL22+αxHp2 has a unique minimum xαX, and this minimum xα is the unique solution of the quasinormal equation A*Axα+αT2xα=A*y.

Proof.

We divide the proof into three steps.

Step 1. The existence of a minimum of Jα(x) is proved. Let {xn}n=1X be a minimizing sequence; that is, Jα(xn)I:=infxXJα(x) as n. We first need to show that {xn}n=1X is a Cauchy sequence. According to the definition of Jα(x), we have (2.3)Jα(xn)+Jα(xm)-2Jα(xn+xm2)=Axn-yL22+αxnHp2+Axm-yL22+αxmHp2-2A(xn+xm2)-yL22-2αxn+xm2Hp2=Axn-y,Axn-yL2+Axm-y,Axm-yL2-2A(xn+xm2)-y,A(xn+xm2)-yL2+αxnHp2+αxmHp2-2αxn+xm2Hp2=12A(xn-xm)L22+α2(2xnHp2+2xmHp2-xn+xmHp2)=12A(xn-xm)L22+α2xn-xmHp2. This implies that Jα(xn)+Jα(xm)2I+(α/2)xn+xmHp2. Since the left-hand side converges to 2I as n,m tend to infinity. This shows that {xn}n=1 is a Cauchy sequence and thus convergent. Let limnxn=xα, noting that xαX. From the continuity of Jα(x), we conclude that Jα(xn)Jα(xα), that is, Jα(xα)=I. This proves the existence of a minimum of Jα(x).

Step 2. The equivalence of the quasinormal equation with the minimization problem for Jα(x) is shown. According to the definition Jα(x) and (2.1), we can obtain the following formula: (2.4)Jα(x)-Jα(xα)=Ax-yL22+αxHp2-Axα-yL22-αxαHp2=Ax-y,Ax-yL2-Axα-y,Axα-yL2+αx,xHp-αxα,xαHp=A(x-xα)L22+2Rex-xα,A*(Axα-y)+αT2xαL2+αT(x-xα)L22, for all xX. If xα satisfies A*Axα+αT2xα=A*y, then Jα(x)-Jα(xα)=A(x-xα)L22+αT(x-xα)L220, that is, xα minimizes Jα(x).

Conversely, if xα minimizes Jα(x), then we substitute x-xα=tξ for any t>0 and ξX, and then we can arrive at (2.5)2tReξ,A*(Axα-y)+αT2xαL2+t2(AξL22+αTξL22)0. Dividing both sides of the above inequality by t>0 and taking t0, we get (2.6)Reξ,A*(Axα-y)+αT2xαL20, for all ξX. This implies that A*(Axα-y)+αT2xα=0. It follows that xα solves the quasinormal equation. From this, the equivalence of the quasinormal equation with the minimization problem for Jα(x) is shown exactly.

Step 3. We show that the operator A*A+αT2 is one-one for every α>0. Let (A*A+αT2)x=0. Multiplication by x yields Ax,AxL2+αTx,TxL2=0, that is, x=0.

3. A Tikhonov-Type Regularization Method

In this section, we first present a Tikhonov-type regularization method to obtain the approximate solution of (1.2) and then consider an a priori strategy and a posteriori choice rule to find the regularization parameter. Under each choice of the regularization parameter, the corresponding estimate can be obtained.

Since (1.2) is an ill-posed problem, we give its regularized solution fα,δ(x) which minimizes the Tiknonov functional (3.1)Jα(f(x)):=Kf-gδL22+αfHp2, where the operator K is defined as in (1.10), and α>0 is a regularization parameter.

According to Lemma 2.1, this minimum fα,δ(x) is the unique solution of the quasinormal equation K*Kfα,δ(x)+αT2fα,δ(x)=K*gδ(x), that is, fα,δ(x)=(K*K+αT2)-1K*gδ(x). Because K is a linear self-adjoint operator, that is, K*=K, we have the equivalent form of fα,δ(x) as (3.2)fα,δ(x)=(K2+αT2)-1Kgδ(x). Further, the function fα,δ(x) can be reduced to (3.3)fα,δ(x)=-n=1(n2+k2)/(1-e-n2+k2)1+α((1+n2)p((n2+k2)/(1-e-n2+k2)))2gnδXn(x).

Now we are ready to formulate the main results of this paper. Before proceeding, the following lemmas are needed.

Lemma 3.1.

For any nN+, k>0, it holds n2+k2(n2+k2)/(1-e-n2+k2)1+n2+k2.

Proof.

The proof is elementary and is omitted.

Lemma 3.2.

For k>0, p0, hL2(0,π) and the operator K defined in (1.10), one has (3.4)K-1hL2(1+k2)p/(p+2)hL2p/(p+2)K-1hHp2/(p+2).

Proof.

By the Hölder inequality and Lemma 3.1, we have (3.5)K-1hL22=n=1(n2+k21-e-n2+k2|hn|)2=n=1(n2+k21-e-n2+k2)2|hn|4/(p+2)|hn|2p/(p+2)(n=1((n2+k21-e-n2+k2)2|hn|4/(p+2))(p+2)/2)2/(p+2)×(n=1(|hn|2p/(p+2))(p+2)/p)p/(p+2)=(n=1(n2+k21-e-n2+k2)p+2|hn|2)2/(p+2)hL22p/(p+2)=(n=1(n2+k21-e-n2+k2)p(n2+k21-e-n2+k2|hn|)2)2/(p+2)hL22p/(p+2)(n=1(1+n2)p(1+n2+k21+n2)p(n2+k21-e-n2+k2|hn|)2)2/(p+2)hL22p/(p+2)(1+k2)2p/(p+2)hL22p/(p+2)K-1hHp4/(p+2). The proof is completed.

In the following we give the corresponding convergence results for an a priori choice rule and an a posteriori choice rule.

3.1. An A Priori Choice Rule

Choose the regularization parameter α1 as (3.6)α1=(δE)2.

The next theorem shows that the choice (3.6) is valid under suitable assumptions.

Theorem 3.3.

Let fα1,δ(x) be the minimizer of Jα1(f(x)) defined by (3.1) and f(x) be the exact solution of (1.2), and let assumptions (A), (B), and (C) hold. If α1 is chosen by (3.6), then fα1,δ(x) is convergent to the exact solution f(x) as the noise level δ tends to zero. Furthermore, one has the following estimate: (3.7)fα1,δ-fL2(2+1)(1+k2)p/(p+2)δp/(p+2)E2/(p+2).

Proof.

Since fα1,δ(x) is the minimizer of Jα1(f(x)) defined by (3.1), we can obtain (3.8)fα1,δHp21α1Jα1(fα1,δ(x))1α1Jα1(f(x))2E2,Kfα1,δ-gδL22Jα1(fα1,δ(x))Jα1(f(x))2δ2. Furthermore, we get (3.9)fα1,δ-fHpfα1,δHp+fHp(2+1)E,Kfα1,δ-gL2Kfα1,δ-gδL2+gδ-gL2(2+1)δ. By Lemma 3.2, we have (3.10)fα1,δ-fL2=K-1(Kfα1,δ-g)L2(1+k2)p/(p+2)Kfα1,δ-gL2p/(p+2)fα1,δ-fHp2/(p+2)(2+1)(1+k2)p/(p+2)δp/(p+2)E2/(p+2). The proof is completed.

3.2. An A Posteriori Choice Rule

Choose the regularization parameter α2 as the solution of the equation (3.11)Kfα2,δ-gδL2=τδ, where the operator K is defined by (1.10) and τ>1.

In the following theorem, an a posteriori rule based on the discrepancy principle  is considered in the convergence estimate.

Theorem 3.4.

Let fα2,δ(x) be the minimizer of Jα2(f(x)) defined by (3.1) and f(x) be the exact solution of (1.2), and let assumptions (A), (B), and (C) hold. If α2 is chosen as the solution of (3.11), then fα2,δ(x) is convergent to the exact solution f(x) as the noise level δ tends to zero. Furthermore, one has the following estimate: (3.12)fα2,δ-fL22(1+k2)p/(p+2)δp/(p+2)E2/(p+2).

Proof.

Since fα2,δ(x) is the minimizer of Jα2(f(x)) defined by (3.1), we can obtain (3.13)Kfα2,δ-gδL22+α2fα2,δHp2=Jα2(fα2,δ(x))Jα2(f(x))=g-gδL22+α2fHp2. Consequently, it has (3.14)fα2,δHp2fHp2+1α2(g-gδL22-τ2δ2)fHp2+1α2(1-τ2)δ2<fHp2E2. This leads to fα2,δ-fHpfα2,δHp+fHp2E. It follows from Lemma 3.2 that the assertion of this theorem is true.

4. Numerical Tests

In this section, we present an example to illustrate the effectiveness and stability of our proposed method. The numerical results verify the validity of the theoretical results for the two cases of the a priori and a posteriori parameter choice rules.

Substituting (1.8) into (1.9), and then using trapezoid's rule to discretize (1.9) can result in the following discrete form: (4.1)-2πi=1M+1n=1N1-e-n2+k2n2+k2f(xi)sin(nxi)sin(nxj)πM=g(xj), where xi=(i-1)M/π, i=1,2,,M+1, and j=1,2,,M+1.

We conduct two tests, and the tests are performed in the following way: first, from (1.6) and (1.11), we can select the source term f(x)=-(1+k2)sinx and then u(x,y)=(1-e-1+k2y)sinx. Consequently, the data function g(x)=(1-e-1+k2)sinx, and (4.2)fHp=(n=1(1+n2)p|fn|2)1/2=2p/2(1+k2)π/2. We choose E=2p/2(1+k2)π/2. Next, we add a random distributed perturbation to each data function, giving the vector (4.3)gδ=g+εrandn(size(g)). The function randn(·) generates arrays of random numbers whose elements are normally distributed with mean 0 and variance 1. Thus, the total noise level δ can be measured in the sense of root mean square error according to (4.4)δ=gδ-g2:=(1M+1j=1M+1(gδ(xj)-g(xj))2)1/2. And gnδ can be obtained according to (1.12). Our error estimates use the relative error, which is given as follows: (4.5)reff(f(x)):=fα,δ-f2f2, where ·2 is given by (4.4).

Test 1.

In the case of the a priori choice rule, we, respectively, compute   reff(f(x)) with different parameters M, N, p, k, and ε. Tables 1 and 2 show that M and N have small influence on reff(f(x)) when they become larger. So, we always take M=100 and N=10 in this test. Table 3 shows reff(f(x)) for p=0,1/2,1,2,4,8, and  10 with the perturbation ε=0.1,0.01, and 0.001. Table 4 shows reff(f(x)) for k=1,2,4, and  8 with the perturbation ε=0.1,0.01, and  0.001. In conclusion, the regularized solution fα,δ well converges to the exact solution f(x) when ε tends to zero.

Relative errors   reff(f(x)) with ε=0.01, k=2, p=2, and N=10 for different M.

M 10 50 100 200 400 800 1600
reff ( f ( x ) ) 0.0471 0.0141 0.0116 0.0036 0.0039 0.0020 0.0018

Relative errors   reff(f(x)) with ε=0.01, k=2, p=2 and M=100 for different N.

N 1 5 10 20 40 80 100
reff ( f ( x ) ) 3.8681 e - 004 0.0055 0.0040 0.0081 0.0067 0.0054 0.0046

Relative errors   reff(f(x)) with k=2, M=100, and N=10, for different p and ε.

reff ( f ( x ) ) p = 0 p = 1 / 2 p = 1 p = 2 p = 4 p = 8 p = 10
ε = 0.1 0.1660 0.1345 0.0564 0.0241 0.0097 0.0076 0.0223
ε = 0.01 0.0283 0.0308 0.0221 0.0040 0.0063 0.0033 0.0021
ε = 0.001 0.0026 0.0048 0.0030 0.0018 0.0011 1.4765 e - 004 1.3137 e - 004

Relative errors   reff(f(x)) with p=2, M=100, and N=10, for different k and ε.

reff ( f ( x ) ) k = 1 k = 2 k = 4 k = 8
ε = 0.1 0.0412 0.0548 0.0304 0.0245
ε = 0.01 0.0133 0.0174 0.0064 0.0036
ε = 0.001 0.0032 0.0031 8.2509 e - 004 5.2171 e - 004
Test 2.

In the case of the a posteriori choice rule (3.11), by taking τ=1.5, we also give the corresponding results as described in Test 1. The results can be easily seen from Tables 5, 6, 7, and 8.

Relative errors   reff(f(x)) with ε=0.01, k=2, p=2, and N=10 for different M.

M 10 50 100 200 400 800 1600
reff ( f ( x ) ) 0.0364 0.0239 0.0194 0.0093 0.0074 0.0060 0.0045

Relative errors   reff(f(x)) with ε=0.01, k=2, p=2, and M=200 for different N.

N 1 5 10 20 40 80 100
reff ( f ( x ) ) 1.0000 0.0041 0.0071 0.0086 0.0098 0.0094 0.0090

Relative errors reff(f(x)) with k=2, M=200, and N=10, for different p and ε.

reff ( f ( x ) ) p = 0 p = 1 / 2 p = 1 p = 2 p = 4 p = 8 p = 10
ε = 0.1 0.1160 0.1849 0.1986 0.1230 0.0179 0.0170 0.0337
ε = 0.01 0.0292 0.0128 0.0227 0.0026 0.0031 0.0040 5.9243 e - 004
ε = 0.001 0.0017 0.0027 0.0010 0.0025 1.3769 e - 004 0.0028 1.1640 e - 004

Relative errors   reff(f(x)) with p=2, M=200, and N=10, for different k and ε.

reff ( f ( x ) ) k = 1 k = 2 k = 4 k = 8
ε = 0.1 0.0250 0.0190 0.0315 0.0516
ε = 0.01 0.0076 0.0074 0.0030 0.0536
ε = 0.001 0.0047 0.0013 5.6243 e - 004 0.0071

From Tests 1 and 2, we conclude that the proposed regularization method is effective and stable.

5. Conclusion

In this paper, we proposed a Tikhonov-type regularization method to deal with the inverse source identification for the modified Helmholtz equation and set up a theoretical frame to analyze the convergence of such method. For instance, we provided the quasinormal equation to obtain the regularized solution. Moreover, besides the a priori parameter choice rule we studied an a posteriori rule for choosing the regularization parameter. Finally, we presented a numerical example whose results seem to be in excellent agreement with the convergence estimates of the method.

Acknowledgments

The authors would like to thank the referee for his (her) valuable comments and suggestions which improved this work to a great extent. This work was supported by the National Natural Science Foundation of China (NSFC) under Grant 40730424, the National Science and Technology Major Project under Grant 2011ZX05023-005, the National Science and Technology Major Project under Grant 2011ZX05044, the National Basic Research Program of China (973 Program) under Grant 2013CB329402, and the National Natural Science Foundation of China (NSFC) under Grant 11131006.

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