A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

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.


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 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 Mathematical Problems in Engineering 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 : 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 g δ x .
To determine the source term f x , we require the following assumptions: A f x ∈ L 2 0, π and g x ∈ L 2 0, π ; B there exists a relation between the function g x and the measured data g δ x : where • L 2 denotes the norm in the space L 2 0, π and δ > 0 is the noise level.
C The source term f x satisfies the a priori bound where E is a positive constant, and • H p denotes the norm in Sobolev space H p 0, π which is defined by 31 as follows: We can refer to 31 for the details of the Sobolev space H p 0, π .Using the separation of variables, we can obtain the explicit solution of the modified Helmholtz equation: where is an orthogonal basis in L 2 0, π , and According to 1.6 and the supplementary condition u x, 1 g x , we have 1.9 Based on this relation, we can define an operator K on the space L 2 0, π as 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 Note that the factor n 2 k 2 / 1 − e − √ n 2 k 2 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.

Preparation
In this section, we give an auxiliary result which will be used in this paper.
We first define an operator T on L 2 0, π as follows: 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 H p 0, π and L 2 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 has a unique minimum x α ∈ X, and this minimum x α is the unique solution of the quasinormal equation A * Ax α αT 2 x α A * y.
Proof.We divide the proof into three steps.
Step 1.The existence of a minimum of J α x is proved.Let {x n } ∞ n 1 ⊂ X be a minimizing sequence; that is, J α x n → I : inf x∈X J α x as n → ∞.We first need to show that {x n } ∞ n 1 ⊂ X is a Cauchy sequence.According to the definition of J α x , we have Since the left-hand side converges to 2I as n, m tend to infinity.This shows that {x n } ∞ n 1 is a Cauchy sequence and thus convergent.Let lim n → ∞ x n x α , noting that x α ∈ X.From the continuity of J α x , we conclude that J α x n → J α x α , that is, J α x α I.This proves the existence of a minimum of J α x .
Step 2. The equivalence of the quasinormal equation with the minimization problem for J α x is shown.According to the definition J α x and 2.1 , we can obtain the following formula: tξ for any t > 0 and ξ ∈ X, and then we can arrive at Dividing both sides of the above inequality by t > 0 and taking t → 0, we get Re ξ, A * Ax α − y αT 2 x α L 2 ≥ 0, 2.6 for all ξ ∈ X.This implies that A * Ax α − y αT 2 x α 0. It follows that x α solves the quasinormal equation.From this, the equivalence of the quasinormal equation with the minimization problem for J α x is shown exactly.
Step 3. We show that the operator A * A αT 2 is one-one for every α > 0. Let A * A αT 2 x 0. Multiplication by x yields Ax, Ax L 2 α Tx, Tx L 2 0, that is, x 0.

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 f α,δ x which minimizes the Tiknonov functional where the operator K is defined as in 1.10 , and α > 0 is a regularization parameter.According to Lemma 2.1, this minimum f α,δ x is the unique solution of the quasinormal equation K * Kf α,δ x αT 2 f α,δ x K * g δ x , that is, f α,δ x K * K αT 2 −1 K * g δ x .Because K is a linear self-adjoint operator, that is, K * K, we have the equivalent form of f α,δ x as Further, the function f α,δ x can be reduced to Now we are ready to formulate the main results of this paper.Before proceeding, the following lemmas are needed.

Lemma 3.1. For any
Proof.The proof is elementary and is omitted.Lemma 3.2.For k > 0, p ≥ 0, h ∈ L 2 0, π and the operator K defined in 1.10 , one has Proof.By the H ölder inequality and Lemma 3.1, we have

3.5
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.

An A Priori Choice Rule
Choose the regularization parameter α 1 as

3.6
The next theorem shows that the choice 3.6 is valid under suitable assumptions.
Theorem 3.3.Let f α 1 ,δ x be the minimizer of J α 1 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 α 1 is chosen by 3.6 , then f α 1 ,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
Proof.Since f α 1 ,δ x is the minimizer of J α 1 f x defined by 3.1 , we can obtain

3.10
The proof is completed.

An A Posteriori Choice Rule
Choose the regularization parameter α 2 as the solution of the equation 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 f α 2 ,δ x be the minimizer of J α 2 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 α 2 is chosen as the solution of 3.11 , then f α 2 ,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
Proof.Since f α 2 ,δ x is the minimizer of J α 2 f x defined by 3.1 , we can obtain Consequently, it has It follows from Lemma 3.2 that the assertion of this theorem is true.

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: where x i 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 k 2 sin x and then u x, y We choose E 2 p/2 1 k 2 π/2.Next, we add a random distributed perturbation to each data function, giving the vector g δ 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

4.3
And g δ n can be obtained according to 1.12 .Our error estimates use the relative error, which is given as follows: 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 f α,δ well converges to the exact solution f x when ε tends to zero.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.
From Tests 1 and 2, we conclude that the proposed regularization method is effective and stable.

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.

Table 1 :
Relative errors reff f x with ε 0.01, k 2, p 2, and N 10 for different M.

Table 2 :
Relative errors reff f x with ε 0.01, k 2, p 2 and M 100 for different N.

Table 3 :
Relative errors reff f x with k 2, M 100, and N 10, for different p and ε.

Table 4 :
Relative errors reff f x with p 2, M 100, and N 10, for different k and ε.

Table 5 :
Relative errors reff f x with ε 0.01, k 2, p 2, and N 10 for different M.

Table 6 :
Relative errors reff f x with ε 0.01, k 2, p 2, and M 200 for different N.

Table 7 :
Relative errors reff f x with k 2, M 200, and N 10, for different p and ε.

Table 8 :
Relative errors reff f x with p 2, M 200, and N 10, for different k and ε.