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 [1] 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 [4–6], inverse heat conduction [7–11], pollution source identification [12], electromagnetic source identification [13], Stefan design problems [14], sound source reconstruction [15], 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 [18–23].
The main difficulty of inverse source identification problems is that they are typically ill posed in the sense of Hadamard [24]. 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 [25], or a nonseparable type is in the form of a moving front [26], 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, 27–29] 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 [29]):
(1.2)Δu(x,y)-k2u(x,y)=f(x),0<x<π,0<y<∞,u(0,y)=u(π,y)=0,0≤y<∞,u(x,0)=0,0≤x≤π,u(x,y)|y→∞bounded,0≤x≤π,u(x,1)=g(x),0≤x≤π,
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-gδ∥L2≤δ,
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)∥f∥Hp≤E,p≥0,
where E is a positive constant, and ∥·∥Hp denotes the norm in Sobolev space Hp(0,π) which is defined by [31] as follows:
(1.5)∥f(·)∥Hp=(∑n=1∞(1+n2)p|〈f,Xn〉|2)1/2.
We can refer to [31] 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=1∞1-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=2π∫0πf(x)sin(nx)dx.
According to (1.6) and the supplementary condition u(x,1)=g(x), we have
(1.9)g(x)=-∑n=1∞1-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=1∞1-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=1∞n2+k21-e-n2+k2gnXn(x),
with
(1.12)gn=2π∫0πg(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 [29], 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 [27].
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:X→Y be a linear and bounded operator between two Hilbert spaces, and let T be defined as in (2.1), α>0. Then for any x∈X the following Tikhonov functional
(2.2)Jα(x):=∥Ax-y∥L22+α∥x∥Hp2
has a unique minimum xα∈X, 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=1∞⊂X be a minimizing sequence; that is, Jα(xn)→I:=infx∈XJα(x) as n→∞. We first need to show that {xn}n=1∞⊂X is a Cauchy sequence. According to the definition of Jα(x), we have
(2.3)Jα(xn)+Jα(xm)-2Jα(xn+xm2)=∥Axn-y∥L22+α∥xn∥Hp2+∥Axm-y∥L22+α∥xm∥Hp2-2∥A(xn+xm2)-y∥L22-2α∥xn+xm2∥Hp2=〈Axn-y,Axn-y〉L2+〈Axm-y,Axm-y〉L2-2〈A(xn+xm2)-y,A(xn+xm2)-y〉L2+α∥xn∥Hp2+α∥xm∥Hp2-2α∥xn+xm2∥Hp2=12∥A(xn-xm)∥L22+α2(2∥xn∥Hp2+2∥xm∥Hp2-∥xn+xm∥Hp2)=12∥A(xn-xm)∥L22+α2∥xn-xm∥Hp2.
This implies that Jα(xn)+Jα(xm)≥2I+(α/2)∥xn+xm∥Hp2. 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 limn→∞xn=xα, noting that xα∈X. 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-y∥L22+α∥x∥Hp2-∥Axα-y∥L22-α∥xα∥Hp2=〈Ax-y,Ax-y〉L2-〈Axα-y,Axα-y〉L2+α〈x,x〉Hp-α〈xα,xα〉Hp=∥A(x-xα)∥L22+2Re〈x-xα,A*(Axα-y)+αT2xα〉L2+α∥T(x-xα)∥L22,
for all x∈X. If xα satisfies A*Axα+αT2xα=A*y, then Jα(x)-Jα(xα)=∥A(x-xα)∥L22+α∥T(x-xα)∥L22≥0, that is, xα minimizes Jα(x).
Conversely, if xα minimizes Jα(x), then we substitute x-xα=tξ for any t>0 and ξ∈X, and then we can arrive at
(2.5)2tRe〈ξ,A*(Axα-y)+αT2xα〉L2+t2(∥Aξ∥L22+α∥Tξ∥L22)≥0.
Dividing both sides of the above inequality by t>0 and taking t→0, we get
(2.6)Re〈ξ,A*(Axα-y)+αT2xα〉L2≥0,
for all ξ∈X. 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,Ax〉L2+α〈Tx,Tx〉L2=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-gδ∥L22+α∥f∥Hp2,
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 n∈N+, 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, p≥0, h∈L2(0,π) and the operator K defined in (1.10), one has
(3.4)∥K-1h∥L2≤(1+k2)p/(p+2)∥h∥L2p/(p+2)∥K-1h∥Hp2/(p+2).
Proof.
By the Hölder inequality and Lemma 3.1, we have
(3.5)∥K-1h∥L22=∑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)∥h∥L22p/(p+2)=(∑n=1∞(n2+k21-e-n2+k2)p(n2+k21-e-n2+k2|hn|)2)2/(p+2)∥h∥L22p/(p+2)≤(∑n=1∞(1+n2)p(1+n2+k21+n2)p(n2+k21-e-n2+k2|hn|)2)2/(p+2)∥h∥L22p/(p+2)≤(1+k2)2p/(p+2)∥h∥L22p/(p+2)∥K-1h∥Hp4/(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-f∥L2≤(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,δ∥Hp2≤1α1Jα1(fα1,δ(x))≤1α1Jα1(f(x))≤2E2,∥Kfα1,δ-gδ∥L22≤Jα1(fα1,δ(x))≤Jα1(f(x))≤2δ2.
Furthermore, we get
(3.9)∥fα1,δ-f∥Hp≤∥fα1,δ∥Hp+∥f∥Hp≤(2+1)E,∥Kfα1,δ-g∥L2≤∥Kfα1,δ-gδ∥L2+∥gδ-g∥L2≤(2+1)δ.
By Lemma 3.2, we have
(3.10)∥fα1,δ-f∥L2=∥K-1(Kfα1,δ-g)∥L2≤(1+k2)p/(p+2)∥Kfα1,δ-g∥L2p/(p+2)∥fα1,δ-f∥Hp2/(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-gδ∥L2=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 [27] 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-f∥L2≤2(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-gδ∥L22+α2∥fα2,δ∥Hp2=Jα2(fα2,δ(x))≤Jα2(f(x))=∥g-gδ∥L22+α2∥f∥Hp2.
Consequently, it has
(3.14)∥fα2,δ∥Hp2≤∥f∥Hp2+1α2(∥g-gδ∥L22-τ2δ2)≤∥f∥Hp2+1α2(1-τ2)δ2<∥f∥Hp2≤E2.
This leads to ∥fα2,δ-f∥Hp≤∥fα2,δ∥Hp+∥f∥Hp≤2E. 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)-2π∑i=1M+1∑n=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)∥f∥Hp=(∑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-g∥2:=(1M+1∑j=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-f∥2∥f∥2,
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, and10 with the perturbation ε=0.1,0.01, and 0.001. Table 4 shows reff(f(x)) for k=1,2,4, and8 with the perturbation ε=0.1,0.01, and0.001. 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))
3.8681e-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
1.4765e-004
1.3137e-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
8.2509e-004
5.2171e-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
5.9243e-004
ε=0.001
0.0017
0.0027
0.0010
0.0025
1.3769e-004
0.0028
1.1640e-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
5.6243e-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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