A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.


Introduction
The inversion problem on mathematical physics equations is an important branch in mathematics.The inversion of surface parameters in remote sensing science, the invention and application of CT in medical imaging, the reconstruction of optical signals and the geomorphological exploration in geological exploration are all related to such inversion problems [1].
There are many research methods and results for the Helmholtz equation for a positive wave number.The Helmholtz equation of a pure imaginary wave number is called the modified Helmholtz equation (also known as the Yukawa equation).It usually appears in the semi-implicit time-discrete heat equation and is also used to describe the physical phenomena of wave dispersion and diffusion [2].A number of numerical solutions in the direct problem for the modified Helmholtz equation have been proposed [3,4], however, its inverse problem is severely ill-posed or improperly-posed in the viewpoint of Hadamard [5], the Cauchy problem suffers from the instability of the solution in the sense that a minor disturbance in the input data may cause a tremendous deviation in the solution [6].To establish an accurate, stable, reliable and fast numerical algorithm for the Cauchy problem is a considerably interesting topic.It is impossible to solve this problem by using classical numerical methods, such as the finite element method (FEM), finite difference method (FDM) and finite volume method (FVM).It requires special techniques, and some different approaches had been given of an account of on the published literature, for instance, the Landweber method with boundary element method (BEM) [7], the conjugate gradient method [8], the method of fundamental solutions (MFS) [9], the Fourier regularization method [10], the truncation method [11] and the mollification method [12,13].
The kernel functions, such as Féjer kernel, Weierstrass kernel, Bessel-MacDonald kernel, de la Vallée Poussin kernel, Dirichlet and Krein kernel have a wide range of applications [12].Manselli, Miller [13] and Murio [14] adopted the mollification method with the Weierstrass kernel to construct regularization operators to solve some inverse problems, but their methods were suitable for the case of Hilbert space L 2 (R), furthermore they could not find appropriate regularization parameters.Hào [15] generalized their works, not only in Hilbert spaces, but also in Banach spaces with de la Vallée Poussin kernel and Dirichlet kernel.He applied the mollification method to concrete problems, such as the Cauchy problem of the Laplace equation, numerical differentiation and some parabolic equations.In recent years, inspired by Murio's work [16], there are some research results for the mollification method with Gaussian kernel to solve Cauchy problem of elliptic equations [17][18][19].
In this paper, a mollification regular method with the de la Vallée Poussin kernel is introduced to solve the Cauchy problem for a Helmholtz-type equation; our approximation is to transform the ill-posed problem into a well-posed problem by convoluting the de la Vallée Poussin function and the measured data.We consider the following two problems: and where ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 is a two-dimensional Laplace operator, f (y), g(y) are given vectors in L 2 (R), and h 1 (y), h 1 (y) ∈ L 2 (R) are unknown vectors.The constant k (k > 0) is the wave number.
Let w = u + v, where u and v are solution of problems (1) and (2), respectively.Then w is the solution of the following Cauchy problem with the inhomogeneous Neumann boundary condition: where h(y) = h 1 (y) + h 2 (y).
We assume that all the functions involved are L 2 functions in (−∞, ∞), Additionally suppose that input functions f (y), g(y) and its measurement data f δ (y), g δ (y) where δ > 0 denotes noise level, and • denotes the L 2 −norm.
Assume there exits a constant E > 0, such that the following a priori bound holds: The rest of this paper is organized as follows.In Section 2, we illuminate the ill-posed nature of problems (1) and (2), the de la Vallée Poussin kernel and some properties are presented to obtain the regularization solution.In Section 3, some error estimates are given for 0 < x < d and at the boundary x = d under the suitable choices of regularization parameter.Section 4 is the numerical aspect of our proposed algorithm.Some conclusions are given in Section 5.

Description of the Problem and Mollification Method
Let us introduce the Sobolev space H r (R) [15] with r ≥ 0, and if r = 0, then with the Fourier transform Additionally, the inverse Fourier transform for the variable ξ ∈ R In this paper, we denote It is well known that [16] (ϕ and and the Parseval equality [15]

Ill-posed Analysis
Applying the Fourier transform to problems (1) and (2).with respect to the variable y, we obtain the following problems: and The solution of problem (11) The solution of problem ( 12) is or equivalently, e iξy dξ.
Apparently, the factors cosh(x are unbounded with respect to variable ξ, a small perturbation in the measured data ĝ(ξ) and h(ξ) may arouse a tremendously large error in the solutions u(x, y) and v(x, y), respectively.Therefore, problems (1), ( 2) and ( 3) are severely ill-posed [5].

Mollification Method
This paper is devoted to establishing a mollification method, constructing mollification operator by convolution with the de la Vallée Poussin kernel and measurement data, as thus the ill-posed problems are transformed into well-posed problems.
(1) V α (t) is an exponential type entire function of degree 2α relative variable t, bounded and summable on R; (2) Vα is the Fourier transform of V α (t) and satisfies We define the operator T α by T α : R → R The Cauchy problems ( 1) and ( 2) can be stabilized, if instead of attempting to find the values of the function f δ (y), g δ (y), we shall reconstruct the δ−mollification of the function f δ (y), g δ (y), given by (T α f δ )(y), (T α g δ )(y).We have the following problems with the mollified data: and where u α,δ and v α,δ denote the solution of problems ( 13) and ( 14), respectively.The solution of problem ( 13) is The solution of problem ( 14) is e iξy dξ.
According to the property (3) of kernel function V α , we have From (9) and property (4) of V α , we get Remark 1. Assumption that condition (4) is valid, when α ≥ 1, we have
From inequality, sinh(x can be arrived at.
In the following, we will give error estimate for u − u α,δ , v − v α,δ and w − w α,δ in 0 < x < d and at boundary x = d, respectively.The convergence results will be obtained while we choose a suitable regular parameter α.

Approximation Theorems
In this section, we shall give the stable estimates of the proposed regularization method for the case of 0 < x < d.Theorem 1.Let u(x, y) be the exact solution of problem (1) with the exact input data f (y), and let u α,δ (x, y) be the regularized solution of problem (13) with the noisy data f δ (y).Assume that conditions (4) and ( 5) hold, we have the following estimate: Furthermore, if we select regular parameter α as α = 1 2d ln(E/δ), ( 16) Proof.Suppose that conditions (4) and ( 5) hold, using Parseval formula (10), we have where According to Minkowski inequality, there is where, Therefore, we have From items ( 3) and ( 4) of Lemma 1, we have Using inequality Choosing the parameter α as ( 16), then (17) holds.
Similarly, we have the following error estimate for problem (2).
Theorem 2. Let v(x, y) be the exact solution of problem ( 2) with the exact input data g(y), and let v α,δ (x, y) be the regularized solution of problem ( 14) with the noisy data g δ (y).Assume that conditions (4) and ( 5) hold, we have the following estimate: Furthermore, if we select regular parameter α as ( 16), then Moreover, according to the results of Theorems 1 and 2 and Minkowski inequality, we get the Theorem 3 as follows.
Theorem 3. Let w(x, y) be the exact solution of problem (3) with the exact input data f (y), g(y), and let w α,δ (x, y) be its approximate solution with the noisy data f δ (y), g δ (y).Assume that conditions (4) and ( 5) hold, we have the following estimate: Furthermore, if we select regular parameter α as ( 16), then

Approximation Estimate at Boundary
Note that the error estimates in above section only solve our problems for 0 < x < d and do not give any useful information at x = d.In order to obtain the stability estimates of problems ( 1) and ( 2) at x = d, we need a stronger a priori assumption instead of (5): where constant E r > 0 dependents only on r.
Theorem 4. Let u(d, y) be the solution of the Cauchy problem (1) and u α,δ (d, y) be solution of modified problem (13) at x = d.Suppose that conditions (4) and ( 22) hold, we have there is Proof.From equality (10), adoption similar analysis method with Theorem 1, we have where Applying (4) of Lemma 1, we have Taking regularization parameter α as (24), then (25) holds.
Similarly, combining (2) of Lemma 1, we have error estimation of problem (2) at boundary x = d.Theorem 5. Let v(d, y) be the solution of the Cauchy problem (2) and v α,δ (d, y) be the solution of modified problem ( 14) at x = d.Suppose that conditions (4) and ( 22) hold, we have If we select α as (24), then According the results of Theorem 4, Theorem 5 and Minkowski inequality, we get the following approximate estimate of problem (3).Theorem 6.Let w(d, y) be the solution of the Cauchy problem (3) and w α,δ (d, y) be its approximation solution at x = d.Suppose that conditions (4) and ( 22) hold, we have If we select α as (24), then

Numerical Aspect
In this section, in order to test the feasibility and stability of our method, two numerical results are proposed.Numerical experiments are performed by MATLAB R2014b (MathWorks, Natick, MA, USA).
In the numerical examples, we select the discrete interval as −2π ≤ y ≤ 2π, the measurement data f δ (y) and g δ (y) are obtained as follows Function 'rand(•)' generates arrays of random numbers whose elements are normally disturbed with mean 0, variance 2 = 1.
The error level δ is given by In numerical examples, we need to take the discrete Fourier transform of the data vector f δ as follows And the discrete Fourier transform of ûα,δ , where , and N is the total test points at y-axis.To measure the accuracy of the numerical solution u α,δ , we define relative error rel(u) between the exact solution u and approximate solution u α,δ : In following numerical experiments, we always take N = 30, d = 1, and fix the reconstructed position x = 0.2, the a priori mollification parameter α is determined by (16).
Example 1.It's easy to verify that function v(x, y) = sin(y/2) sinh(x is the exact solution of problem (2), where g(y) = sin(y).
To verify the stability of our method, different noisy levels for δ = 1 × 10 −3 , 1 × 10 −4 , 1 × 10 −5 , 1 × 10 −6 are presented, respectively.Tables 1 and 2 show the results associated with different error levels δ of Example 1 and 2. Note that the relative error depends not only on error level δ but also on wave number k.However, due to the characteristic of the selected parametric formula, the numerical results show that the optimal range of error level within 10 −5 and 10 −3 .Figure 1 shows the reconstructed solution and exact solution for Example 1 corresponding to noise levels δ = 1 × 10 −6 , δ = 1 × 10 −3 with k = 0.8 and k = 10, respectively.Figure 2 shows the reconstructed solution and exact solution for Example 2 corresponding to noise levels δ = 1 × 10 −3 with k = 0.8.and k = 5.Note that the proposed method is effective and stable to noisy data.Remark 2. In above two examples, the value of wave number k which we take is relatively small.In fact, when k ∈ [0, 10], the result is still valid, when k > 10, the relative error will gradually increase, and the fitting effect will become more and more undesirability.Moreover, if we take N to be odd, there will be singularities, in this case, instead of [−2π, 2π], we consider interval [−2π − eps, 2π + eps], where eps = 10 −8 is the number of disturbance.

Table 2 .
The relative error rel(u) of Example 2.