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

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 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: Δ𝑢𝑥,𝑦+𝑘²𝑢()𝑢𝑦𝑢𝑥,𝑦=0,𝑥∈0,1,𝑦∈ℝ,0,𝑦=𝜑,𝑦∈ℝ,_𝑥0,𝑦=0,𝑦∈ℝ, (1) 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 𝜑^𝛿(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 𝑔(𝜉)=1√2𝜋^∞−∞𝑒^{−𝑖𝑦𝜉}𝑔𝑦𝑑𝑦,𝑖=√−1. (2) The functions 𝜑(𝑦),𝜑^𝛿(𝑦)∈𝐿²(ℝ) are the exact and measured data for problem (1), respectively, and satisfy 𝜑^𝛿(⋅)−𝜑(⋅)≤𝛿, (3) where ‖⋅‖ denotes the 𝐿²-norm and the constant 𝛿>0 is the noise level. Assume that 𝑢(𝑥,⋅)∈𝐿²(ℝ) for all 0≤𝑥<1 and there is the following a priori bound; ‖𝑢(1,⋅)‖≤𝐸, (4) where 𝐸 is a positive constant.

It is easy to know that for problem (1), 𝑢(𝑥,𝜉)=cosh𝑥𝜉²−𝑘²𝜑(𝜉), (5) and equivalently, 𝑢=1𝑥,𝑦√2𝜋^∞−∞𝑒^𝑖𝑦𝜉cosh𝑥𝜉²−𝑘²𝜑(𝜉)𝑑𝜉. (6)

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 𝜑^𝛿(𝑥) is severely ill-posed.

For simplicity [15], we decompose ℝ into the following parts 𝐼 and 𝑊, where: 𝐼∶=𝜉∈ℝ,𝜉≥𝑘,𝑊∶=𝜉∈ℝ,𝜉≤𝑘; (7) then 𝐿²(ℝ)=𝐿²(𝐼)⊕𝐿²(𝑊).

For 𝜉∈𝑊, we can take the regularization approximation solution in the frequency domain as 𝑢^𝛿_𝑚(𝑥,𝜉)=cos𝑥𝑘²−𝜉²𝜑^𝛿(𝜉). (8)

For 𝜉∈𝐼, we introduce an iteration scheme with the following form: 𝑢^𝛿_𝑚(𝑥,𝜉)=(1−𝜆)𝑢^𝛿_𝑚−1()𝑥𝑥,𝜉+𝜆cosh𝜉²−𝑘²𝜑^𝛿(𝜉),𝑚=1,2,…, (9) where 𝜆=𝜆(𝜉)=𝑒^{−|𝜉|2−𝑘2}<1 plays an important role in the convergence proof; the initial guess is 𝑢^𝛿₀(𝑥,𝜉). By using an elementary calculation for (9), we obtain 𝑢^𝛿_𝑚(𝑥,𝜉)=(1−𝜆)^𝑚𝑢^𝛿₀()+𝑥,𝜉_{𝑚−1𝑖=0}(1−𝜆)^𝑖𝜆cosh𝑥𝜉²−𝑘²𝜑^𝛿(𝜉)=(1−𝜆)^𝑚𝑢^𝛿₀(𝑥,𝜉)+1−(1−𝜆)^𝑚×cosh𝑥𝜉²−𝑘²𝜑^𝛿(𝜉). (10) Therefore, the approximate solution of problem (1) has the following form in the frequency domain: 𝑢^𝛿_𝑚()=𝑥𝑥,𝜉cos𝑘²−𝜉²𝜑^𝛿(𝜉),𝜉()≤𝑘,1−𝜆^𝑚𝑢^𝛿₀()+()𝑥,𝜉1−1−𝜆^𝑚×cosh𝑥𝜉²−𝑘²𝜑^𝛿(𝜉),𝜉≥𝑘, (11) or equivalently, 𝑢^𝛿_𝑚=1𝑥,𝑦√2𝜋^∞−∞𝑒^𝑖𝑥𝜉𝑢^𝛿_𝑚(𝑥,𝜉)𝑑𝜉, (12) where 𝑢^𝛿_𝑚(𝑥,𝜉) is given by (11).

Lemma 1 (see [16]). For 0<𝜆≤1 and 𝑚≥1, the following inequalities hold: (1−𝜆)^𝑚1𝜆≤,()𝑚+11−1−𝜆^𝑚𝜆≤𝑚. (13)

Lemma 2. For 0≤𝜆≤1,0≤𝛼≤1 and 𝑚≥1, Lemma 1 can be strengthened as the following inequalities: (1−𝜆)^𝑚𝜆^𝛼≤(𝑚+1)^−𝛼,1−(1−𝜆)^𝑚𝜆^𝛼≤𝑚^𝛼. (14)(15)

Proof. In fact, using the established results (13), we can get (1−𝜆)^𝑚𝜆^𝛼≤(1−𝜆)^𝑚𝜆^𝛼≤(𝑚+1)^−𝛼,1−(1−𝜆)^𝑚𝜆^𝛼≤1−(1−𝜆)^𝑚𝜆^𝛼≤𝑚^𝛼. (16)

Theorem 3. Let 𝑢(𝑥,𝑦) be the exact solution of problem (1) and 𝑢^𝛿_𝑚(𝑥,𝑦) be its regularized approximation given by (12) with 𝑢^𝛿₀(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: 𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)≤3𝐸^𝑥𝛿^1−𝑥+𝛿,0<𝑥<1. (17)

Proof. Due to the Parseval formula and the triangle inequality, we have 𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)²=𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)²=𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)²_𝐿2(𝑊)+𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)²_𝐿2(𝐼). (18)
Case  1. While 𝜉∈𝑊, combining (3), (6), and (12), we have 𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)_𝐿2(𝑊)=cos𝑥𝑘²−𝜉²𝜑(𝜉)−cos𝑥𝑘²−𝜉²𝜑^𝛿(𝜉)_𝐿2(𝑊)≤𝜑(𝜉)−𝜑^𝛿(𝜉)_𝐿2(𝑊)≤𝜑(𝜉)−𝜑^𝛿(𝜉)≤𝛿. (19)
Case  2. For 𝜉∈𝐼, combining (4), (6), (12), (14), and (15), we have 𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)_𝐿2(𝐼)=cosh𝑥𝜉²−𝑘²𝜑−cosh𝑥𝜉²−𝑘²1−(1−𝜆)^𝑚𝜑^𝛿≤cosh𝑥𝜉²−𝑘²(1−𝜆)^𝑚𝜑+cosh𝑥𝜉²−𝑘²1−(1−𝜆)^𝑚𝜑−𝜑^𝛿=cosh𝑥𝜉²−𝑘²(1−𝜆)^𝑚cosh𝜉²−𝑘²cosh𝜉²−𝑘²𝜑+cosh𝑥𝜉²−𝑘²1−(1−𝜆)^𝑚𝜑^𝛿−𝜑≤𝐸sup_𝜉∈ℝcosh𝑥𝜉²−𝑘²(1−𝜆)^𝑚cosh𝜉²−𝑘²+𝛿sup_𝜉∈ℝcosh𝑥𝜉²−𝑘²1−(1−𝜆)^𝑚≤2𝐸sup_𝜉∈ℝ(1−𝜆)^𝑚𝑒^{(𝑥−1)|𝜉|2−𝑘2}+𝛿sup_𝜉∈ℝ1−(1−𝜆)^𝑚𝑒^{𝑥|𝜉|2−𝑘2}=2𝐸sup_0<𝜆<1𝜆^(1−𝑥)(1−𝜆)^𝑚+𝛿sup_0<𝜆<1𝜆^−𝑥1−(1−𝜆)^𝑚≤2𝐸(𝑚+1)^𝑥−1+𝛿𝑚^𝑥. (20)
Due to 𝑚=⌊𝐸/𝛿⌋, then 𝑚≤𝐸/𝛿 and 𝑚+1≥𝐸/𝛿, therefore, 𝑢(𝑥,⋅)−𝑢^𝛿_𝑚(𝑥,⋅)_𝐿2(𝐼)𝐸≤2𝐸𝛿^𝑥−1𝐸+𝛿𝛿^𝑥=3𝐸^𝑥𝛿^1−𝑥. (21) 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, [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: 𝜑^𝛿−𝑢^𝛿_𝑚(0,⋅)≤𝜏𝛿≤𝜑^𝛿−𝑢^𝛿_𝑚−1(0,⋅), (22) 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^𝛿₀(𝑥,𝜉)=0, thus (22) can be simplified to (1−𝜆)^𝑚𝜑^𝛿_𝐿2(𝐼)≤𝜏𝛿≤(1−𝜆)^𝑚−1𝜑^𝛿_𝐿2(𝐼). (23)

Lemma 5. The following inequality holds: 𝑚≤2𝐸()𝛿.𝜏−1 (24)

Proof. Due to (4) and (14), we know 𝜏𝛿≤𝜑^𝛿−𝑢^𝛿_𝑚−1(0,⋅)=(1−𝜆)^𝑚−1𝜑^𝛿_𝐿2(𝐼)≤(1−𝜆)^𝑚−1𝜑^𝛿−𝜑_𝐿2(𝐼)+(1−𝜆)^𝑚−1𝜑_𝐿2(𝐼)≤𝜑^𝛿−𝜑_𝐿2(𝐼)+(1−𝜆)^𝑚−1cosh𝜉²−𝑘²𝜑cosh𝜉²−𝑘²_𝐿2(𝐼)≤𝛿+2𝐸sup_0<𝜆<1(1−𝜆)^𝑚−1𝜆≤𝛿+2𝐸𝑚, (25) therefore, 𝑚≤2𝐸()𝛿.𝜏−1 (26)

Lemma 6. Setting 𝜔_𝑚(𝑥,⋅)=𝑢(𝑥,⋅)−𝑢_𝑚(𝑥,⋅), then the following inequality holds: 𝜔_𝑚(𝑥,⋅)_𝐿2(𝐼)≤2𝜔_𝑚(1,⋅)^𝑥_𝐿2(𝐼)𝜔_𝑚(0,⋅)_𝐿^1−𝑥_2(𝐼). (27)

Proof. Defining 𝑢_𝑚(𝑥,⋅)=(1−(1−𝜆)^𝑚)cosh(𝑥|𝜉|²−𝑘²)𝜑(𝜉), then we have 𝜔_𝑚(0,⋅)=(1−𝜆)^𝑚𝜔𝜑,_𝑚(1,⋅)=(1−𝜆)^𝑚cosh𝜉²−𝑘²𝜔𝜑._𝑚(𝑥,⋅)²_𝐿2(𝐼)=(1−𝜆)^𝑚cosh𝑥𝜉²−𝑘²𝜑²_𝐿2(𝐼)=_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘22𝑚}cosh𝑥𝜉²−𝑘²𝜑²≤𝑑𝜉_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘22𝑚}𝑒^{𝑥|𝜉|2−𝑘2}𝜑²=𝑑𝜉_𝜉∈𝐼1−𝑒^{|𝜉|2−𝑘22𝑚𝑥}𝑒^{|𝜉|2−𝑘2}𝜑^2𝑥×1−𝑒^{|𝜉|2−𝑘22𝑚(1−𝑥)}𝑒^{|𝜉|2−𝑘2}𝜑^2(1−𝑥)≤𝑑𝜉_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘2𝑚}𝑒^{|𝜉|2−𝑘2}𝜑²𝑑𝜉^𝑥×_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘2𝑚}𝜑²𝑑𝜉^1−𝑥≤4_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘2𝑚}cosh𝜉²−𝑘²𝜑²𝑑𝜉^𝑥×_𝜉∈𝐼1−𝑒^{−|𝜉|2−𝑘2𝑚}𝜑²𝑑𝜉^1−𝑥=4𝜔_𝑚(1,⋅)_𝐿^2𝑥_2(𝐼)𝜔_𝑚(0,⋅)_𝐿^2(1−𝑥)_2(𝐼). (28)

Lemma 7. The following inequality holds: 𝑢_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝐿2(𝐼)≤2(𝜏+1)^1−𝑥𝐸^𝑥𝛿^1−𝑥. (29)

Proof. Due to (3), (4), and (27), we know that 𝑢_𝑚(1,⋅)−𝑢(1,⋅)_𝐿2(𝐼)=(1−𝜆)^𝑚cosh𝜉²−𝑘²𝜑𝑢≤𝐸,_𝑚(0,⋅)−𝑢(0,⋅)_𝐿2(𝐼)=(1−𝜆)^𝑚𝜑_𝐿2(𝐼)≤1−𝑒^{−|𝜉|2−𝑘2𝑚}𝜑−𝜑^𝛿_𝐿2(𝐼)+1−𝑒^{−|𝜉|2−𝑘2𝑚}𝜑^𝛿_𝐿2(𝐼)≤(𝜏+1)𝛿. (30) Combining (30), we have 𝑢_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝐿2(𝐼)≤2(𝜏+1)^1−𝑥𝐸^𝑥𝛿^1−𝑥. (31)

Theorem 8. Let 𝑢(𝑥,𝑦) be the exact solution of problem (1) and 𝑢^𝛿_𝑚(𝑥,𝑦) be its regularization approximation defined by (12) with 𝑢^𝛿₀(𝑥,𝑦)=0. If the a priori bound (4) is valid and the iteration (9) is stopped by the discrepancy principle (22), then 𝑢^𝛿_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)≤𝐶𝐸^𝑦/𝑦0𝛿^{1−(𝑦/𝑦0)}+𝛿, (32) where 𝐶=(2/(𝜏−1))^𝑥+2(𝜏+1)^1−𝑥.

Proof. According to the triangle inequality, (19), (24), and (29), we obtain that 𝑢^𝛿_𝑚()(≤𝑢𝑥,⋅−𝑢𝑥,⋅)_𝛿𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝐿2(𝐼)+𝑢^𝛿_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝐿2(𝑊)≤𝑢^𝛿_𝑚(𝑥,⋅)−𝑢_𝑚(𝑥,⋅)_𝐿2(𝐼)+𝑢_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝐿2(𝐼)+𝛿≤𝛿𝑚^𝑥+2(𝜏+1)^1−𝑥𝐸^𝑥𝛿^1−𝑥+𝛿≤𝛿2𝐸(𝜏−1)𝛿^𝑥+2(𝜏+1)^1−𝑥𝐸^𝑥𝛿^1−𝑥=2+𝛿𝜏−1^𝑥+2(𝜏+1)^1−𝑥𝐸^𝑥𝛿^1−𝑥+𝛿. (33)

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 𝑎≤𝑦≤𝑏,  𝑁 denotes the number of discrete points.

For an exact data function 𝜑(𝑦), its discrete noisy version is 𝜑^𝜖=𝜑+𝜖rand𝑛size𝜑, (34) where 𝜑=𝜑𝑦₁,…,𝜑𝑦_𝑁,𝑦_𝑗()=𝑎+𝑏−𝑎𝑗−1𝜑𝑁+1,𝑗=1,2…,𝑁,𝛿=^𝜖−𝜑_𝑙²∶=1𝑁^𝑁_𝑖=1𝜑^𝜖𝑦_𝑖−𝜑𝑦_𝑖². (35) The function “rand𝑛(⋅)” generates arrays of random numbers whose elements are normally distributed with mean 0, variance 𝜎²=1. The absolute error 𝑒_𝑎(𝑢) and the relative error 𝑒_𝑟(𝑢) are defined by 𝑒_𝑎(𝑢)∶=𝑢^𝛿_𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝑙2,𝑒_𝑟(𝑢)𝑢∶=^𝛿𝑚(𝑥,⋅)−𝑢(𝑥,⋅)_𝑙2‖𝑢(𝑥,⋅)‖_𝑙², (36)(37) respectively.

In the numerical experiment, we compute the approximation 𝑢^𝛿_𝑚(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_(ℝ)≈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^−𝑦2∈𝒮(ℝ), where 𝒮(ℝ) denotes the Schwartz function space, 𝜑(𝜉)∈𝒮(ℝ) 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.

Table 1 

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

Figure 1 is the comparison of a priori and a posteriori parameter choice rules for the exact 𝑢(𝑥,𝑦) and the approximate solution 𝑢^𝛿_𝑘(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_(ℝ), and we can see that the a posteriori rule also works effectively.

(a)

(b)

(c)

(d)

Figure 1 

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 𝑢=1𝑥,𝑦𝜋²sin𝜋𝑘𝑦cosh𝑘√𝜋²−1𝑥 (38) is the exact solution of problem (1) with the Cauchy data 𝑢(0,𝑦)=𝜑(𝑦)=(1/𝜋²)sin(𝜋𝑘𝑦) and 𝑢_𝑥(0,𝑦)=0.

Figure 2 is the comparison of the exact solution 𝑢(𝑥,𝑦) and the approximation 𝑢^𝛿_𝑚(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_(ℝ), and the proposed method works well for the a priori parameter choice rule.

(a)

(b)

(c)

(d)

Figure 2 

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.

(a)

(b)

(c)

(d)

Figure 3 

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 𝑢^𝛿_𝑚(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_(ℝ) as previously mentioned, and the a posteriori rule also works effectively as expected.

(a)

(b)

(c)

(d)

Figure 4 

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 1–4, 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).