Research article

A quasi-boundary method for solving an inverse diffraction problem

  • Received: 17 October 2021 Revised: 18 March 2022 Accepted: 21 March 2022 Published: 06 April 2022
  • MSC : 35R25, 35R30, 47A52

  • In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary z=0, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.

    Citation: Zhenping Li, Xiangtuan Xiong, Jun Li, Jiaqi Hou. A quasi-boundary method for solving an inverse diffraction problem[J]. AIMS Mathematics, 2022, 7(6): 11070-11086. doi: 10.3934/math.2022618

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  • In this paper, we deal with the reconstruction problem of aperture in the plane from their diffraction patterns. The problem is severely ill-posed. The reconstruction solutions of classical Tikhonov method and Fourier truncated method are usually over-smoothing. To overcome this disadvantage of the classical methods, we introduce a quasi-boundary regularization method for stabilizing the problem by adding a-priori assumption on the exact solution. The corresponding error estimate is derived. At the continuation boundary z=0, the error estimate under the a-priori assumption is also proved. In theory without noise, the proposed method has better approximation than the classical Tikhonov method. For illustration, two numerical examples are constructed to demonstrate the feasibility and efficiency of the proposed method.



    The reconstruction problem of aperture in the plane from their diffraction patterns arises in acoustics and optics. Consider the following inverse diffraction problem. Let A be a bounded aperture in an infinite perfectly soft screen which is located in the plane z=0 in R3. A harmonic plane wave with wave-number k propagates along with the positive z direction. It hits the screen and escapes through the aperture A. The measured data at receiving screen z=d>0 is given. The problem is to reconstruct the shape (domain) of the aperture A. By Kirchhoff approximation, the mathematical modeling can be formulated as follows.

    Inverse problem. Let u(x,y,z) be the solution of the following problem:

    uxx+uyy+uzz+k2u(x,y,z)=0,(x,y)R2,z>0, (1.1)
    u(x,y,d)=g(x,y),(x,y)R2, (1.2)
    limrr(uriku)=0,wherer=x2+y2+z2. (1.3)

    It is explained here that "r" always equals "x2+y2+z2" in this article. Here, we wish to find the u(x,y,0)=χA(x,y) from the data g(x,y), where χA(x,y) is the characteristic function of the domain A.

    But we deal with a more general case where the χA(x,y) is replaced by a general function f(x,y) that belongs to L2(R2) (if A is the support of f, this corresponds to assuming an arbitrary incident wave).

    Before investigating the inverse problem, we discuss briefly the forward problem. The corresponding forward problem is as follow.

    Forward problem. Let u(x,y,z) be the solution of the following problem:

    uxx+uyy+uzz+k2u(x,y,z)=0,(x,y)R2,z>0, (1.4)
    u(x,y,0)=f(x,y),(x,y)R2, (1.5)
    limrr(uriku)=0. (1.6)

    We wish to find the u(x,y,z)=g(x,y)L2(R2) from the data f(x,y)L2(R2) for a fixed 0<zd.

    It is well known that if f(x,y) has bounded support A, then the forward problem admits a unique solution u(x,y,z) which is given by

    u(x,y,z)=R2Hz(xμ,yν)f(μ,ν)dμdν:=F(z)f(x,y), (1.7)

    where

    Hz(x,y)=12π(eikτ2+x2+y2τ2+x2+y2)τ|τ=z,

    and F(z):L2(A)L2(R2) is the solution operator.

    The problem can be reposed in the Fourier domain. Let

    ˆh(ξ,η)=12πR2h(x,y)ei(ξx+ηy)dxdy (1.8)

    be the Fourier transform of the function h(x,y)L2(R2). The corresponding inverse Fourier transform of the function ˆh(ξ,η) is given by

    h(x,y)=12πR2ˆh(ξ,η)ei(ξx+ηy)dξdη. (1.9)

    By taking Fourier transform of (1.4)–(1.6) with respect to variables x and y, the solution of forward problem in frequency domain is given by

    ˆu(ξ,η,z)=eza(ξ,η)ˆf(ξ,η)=^F(z)f, (1.10)

    where a(ξ,η)=ξ2+η2k2 if ξ2+η2k2 and a(ξ,η)=ik2ξ2η2 if ξ2+η2<k2.

    From this formula, formally we have the solution for the inverse problem

    ˆf(ξ,η)=eda(ξ,η)ˆg(ξ,η) (1.11)

    and

    ˆu(ξ,η,z)=e(dz)a(ξ,η)ˆg(ξ,η). (1.12)

    Following [1], the plane waves with frequencies ξ2+η2<k are called homogeneous waves and the plane waves with frequencies ξ2+η2>k are called evanescent waves. The reconstruction regions are divided two cases:

    (1) Near-field region: Corresponds to distances d<λ=2πk, and in this case the evanescent waves are important.

    (2) Far-field region: Corresponds to distances d>λ=2πk, and in this case the homogeneous waves are important, and one can assume that the ˆu(ξ,η,d) has a support given by B={(ξ,η):ξ2+η2k2}.

    The problem of inverse diffraction from plane to plane is well-known ill-posed problem in the near-field pattern and in the far-field pattern [2,3,4,5,6,7]. There are some machine learning methods for solving inverse scattering problems conveniently, such as neural network [8,9]. In this paper, we discuss the inverse diffraction problem from the aspect of regularization theory for solving ill-posed problems. To stabilize the ill-posed problem and improve the resolution of the solution, a priori information about unknown solution is introduced [10] necessarily.

    The problem is similar to the Cauchy problem for the Helmholtz equation. Regularization methods for solving Cauchy problem have been suggested by various authors. For instance, the modified Tikhonov regularisation method [11,12,13], the truncation method [14,15], quasi-reversibility method [16,17], mapped regularization methods[18], operator regularization method[19], mollification method[20], fractional Tikhonov method[21], a slow-evolution-from-the-continuation-boundary (SECB) method [22], posteriori regularization[23], Spectral Galerkin method[24], etc.

    A priori information about unknown solution has been proved to be essential in the analysis of ill-posed problems in mathematical physics. Otherwise, without the priori information the convergence rate of the constructed regularization method is arbitrarily slow [25,26,27].

    Although this problem has been investigated by Sondhi [28], Bertero [29], Magnanini [10] and Santosa [30], most of the existing results are devoted to the numerical aspects, e.g., a level set method has been proposed for solving the problem in [30]. In this work, we will focus on the regularization method and the error estimate from the viewpoint of regularization for ill-posed problems. In references [22,31], we studied this problem by slow-evolution-from-the-continuation-boundary (SECB) method and spectral method respectively. Here we propose a quasi-boundary regularization for solving this inverse problem and corresponding error estimates are derived. It is well-known that the classical Tikhonov method has an over-smoothing regularization solution and the truncated Fourier method also has one, therefore we try to give a simple method to overcome this disadvantage. The total variation method [32] is a good alternative for resolve this problem, but it is difficult to treat both numerically and theoretically. In this paper, we find out that the proposed quasi-boundary regularization has better approximation than the classical Tikhonov method in the case of noiseless data theoretically. Although this result doesn't has real meaning in practice, the quasi-boundary method has its advantage. The proposed method could give better approximation by using few cost when we want to reconstruct the non-smooth exact object. The numerical examples show the regularization method is effective and support the results.

    The paper is organized as follow. In Section 2, a quasi-boundary regularization method is presented and the error bounds are proved; in Section 3, a comparison between the classical Tikhonov method and the quasi-boundary method is given; in Section 4, some numerical results are reported in order to show that the quasi-boundary method is more effective for the inverse diffraction problem.

    Let g(x,y) denotes the exact data, and we would actually have noisy data function gδ(x,y)L2(R2), for which

    gδ(x,y)g(x,y)δ, (2.1)

    where the constant δ>0 represents a bound on the measured error, denotes the L2norm throughout this paper. An a-priori knowledge about the true solution is an essential element in the successful computation of ill-posed inverse problems [25]. We assume there exists a constant E>0, such that the exact solution u(,,0):=f(,) satisfies

    u(,,0)pE, (2.2)

    where p denotes the norm of Sobolev space Hp(R2).

    In this section, we follow the idea from Ames et al. [33] where they used a quasi-boundary method (or so-called modified boundary method) for solving a classical backward heat equation. Let us consider the problem with noisy data for the current inverse problem:

    uxx+uyy+uzz+k2u(x,y,z)=0,(x,y)R2,z>0, (2.3)
    u(x,y,d)=gδ(x,y),(x,y)R2, (2.4)
    limrr(uriku)=0. (2.5)

    We try to use a quasi-boundary regularization method to solve the problem (2.3)–(2.5), i.e., let us consider the following modified boundary problem:

    (vαδ)xx+(vαδ)yy+(vαδ)zz+k2(vαδ)(x,y,z)=0,(x,y)R2,z>0, (2.6)
    α(vαδ)(x,y,0)+(vαδ)(x,y,d)=gδ(x,y),(x,y)R2, (2.7)
    limrr((vαδ)rik(vαδ))=0, (2.8)

    where α>0 is a small parameter.

    By the technique of Fourier transform, we can get the solution of the modified boundary problem in the frequency domain:

    ^(vαδ)(ξ,η,z)=e(dz)a(ξ,η)1+αeda(ξ,η)^gδ(ξ,η), (2.9)

    where a(ξ,η) is the same as in (1.10).

    Proposition 2.1. Let gδL2(R2) and make the solution of problem (2.3)–(2.5) exist, then the solution of problem (2.6)–(2.8) approximates the solution of problem (2.3)–(2.5) in L2-norm when α0.

    Proof. The solution of (2.3)–(2.5) is given by

    ^uδ(ξ,η,z)=e(dz)a(ξ,η)^gδ(ξ,η).

    From (2.9) we can easily see that as α0, ^vαδ^uδ and ^vαδ^uδ0, uniformly in z.

    Lemma 2.1. If 0z<d, 0<α<1, then

    supξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|αzd1. (2.10)
    supξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|11α. (2.11)

    Proof. If ξ2+η2k2, by inequality (see Lemma 3.1 in [16])

    sups>0esz1+αesdαzd, (2.12)

    we have

    supξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|αdzd=αzd1.

    If ξ2+η2<k2, we have

    supξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|11|αeidk2ξ2η2|=11α.

    Proposition 2.2. Let gδL2(R2), then the solution of problem (2.6)–(2.8) continuously depends on the data in L2-norm if the parameter α is selected appropriately.

    Proof. For problem (2.6)–(2.8), we have the following conclusion: If any two functions gδ,1 and gδ,2 satisfy gδ,1gδ,2ε, let vα,1δ and vα,2δ be the corresponding solutions, respectively, setting α=O(ε), then vα,1δvα,2δ0, as ε0. In fact, by Parseval's identity and Lemma 2.1, we have

    vα,1δvα,2δ=^vα,1δ^vα,2δ=e(dz)a(ξ,η)1+αeda(ξ,η)(^gδ,1^gδ,2)=(ξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2(^gδ,1^gδ,2)|2dξdη+ξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2(^gδ,1^gδ,2)|2dξdη)1/2supξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|^gδ,1^gδ,2+supξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|^gδ,1^gδ,2supξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|ε+supξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|εαzd1ε+ε1α. (2.13)

    Hence if α=O(ε), then

    vα,1δvα,2δ0,for ε0.

    For the inverse problem, we can obtain the error estimate between the regularized solution vαδ(x,y,z) and the exact solution u(x,y,z).

    Theorem 2.1. Suppose u(x,y,z) is the solution of the problem (1.1)–(1.3) with the exact data g(x,y)L2(R2) and vαδ(x,y,z) is the regularization solution whose Fourier transform is given by (2.9) with the noisy data gδ(x,y)L2(R2), let (2.1) and the a-priori condition (2.2) hold with p=0. If α=δE, then for a fixed 0<z<d we have the error estimate

    vαδ(,,z)u(,,z)2δzdE1zd(1+o(1)),forδ0. (2.14)

    Proof. We take two steps to prove it.

    Step I: Convergence. We need to prove the regularization solution vα whose Fourier transform is given by (2.9) approaches the exact solution u with the same exact data g.

    By Parseval's identity, we have

    vα(,,z)u(,,z)=^vα(,,z)ˆu(,,z)=e(dz)a(ξ,η)1+αeda(ξ,η)ˆg(ξ,η)e(dz)a(ξ,η)ˆg(ξ,η)=αe(2dz)a(ξ,η)1+αeda(ξ,η)ˆg(ξ,η). (2.15)

    From (1.12), there holds

    ˆg(ξ,η)=ˆu(ξ,η,0)eda(ξ,η).

    Therefore,

    ^vα(,,z)ˆu(,,z)=αe(dz)a(ξ,η)1+αeda(ξ,η)ˆu(ξ,η,0)=α(ξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2ˆu(ξ,η,0)|2dξdη+ξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2ˆu(ξ,η,0)|2dξdη)1/2αsupξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|((ξ,η)R2|ˆu(ξ,η,0)|2dξdη)1/2+αsupξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|((ξ,η)R2|ˆu(ξ,η,0)|2dξdη)1/2. (2.16)

    By the a-priori assumption (2.2) and Lemma 2.1, (2.16) yields

    ^vα(,,z)ˆu(,,z)αEαzd1+αE1α.

    If α=δ/E, we have

    ^vα(,,z)ˆu(,,z)δzdE1zd+δ1δ/E=δzdE1zd(1+o(1)),forδ0. (2.17)

    Step II: Stability. We now prove that the regularization solution is dependent continuously on the data. Using the Parseval's equality and Lemma 2.1, we obtain

    vαδ(,,z)vα(,,z)=^vαδ(,,z)^vα(,,z)=e(dz)a(ξ,η)1+αeda(ξ,η)(^gδˆg)=(ξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2(^gδˆg)|2dξdη+ξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2(^gδˆg)|2dξdη)1/2supξ2+η2k2|e(dz)ξ2+η2k21+αedξ2+η2k2|δ+supξ2+η2<k2|ei(dz)k2ξ2η21+αeidk2ξ2η2|δαzd1δ+δ1α.

    Noting α=δ/E, we have

    ^vαδ(,,z)^vα(,,z)δzdE1zd+δ1δ/E=δzdE1zd(1+o(1)),forδ0. (2.18)

    The conclusion of the theorem now follows immediately by using the triangle inequalities (2.17) and (2.18).

    We find that the error estimate (2.14) is not valid for the location at z=0. This is common in ill-posed problems. If a stronger a-priori condition is added, for example, u(,,0)pE, with p>0, where p denotes the norm of Sobolev space Hp(R2), then the convergence order is logarithmic.

    Theorem 2.2. Suppose that f(x,y):=u(x,y,0) is the exact solution with exact data g(x,y)L2(R2), (2.9) is the regularization solution with z=0 and noisy data gδ(x,y)L2(R2), respectively, let (2.1) and (2.2) hold with p>0, if α=(δE)r with 0<r<1, then we have for p>0,

    vαδ(,,0)u(,,0)O((lnEδ)p),forδ0. (2.19)

    Proof. (I) (Stability) From Parseval's identity and Lemma 2.1, we have

    vαδ(,,0)vα(,,0)=ˆvαδ(,,0)ˆvα(,,0)=eda(ξ,η)1+αeda(ξ,η)ˆgδ(ξ)eda(ξ,η)1+αeda(ξ,η)ˆg(ξ)=eda(ξ,η)1+αeda(ξ,η)(ˆgδ(ξ)ˆg(ξ))=(ξ2+η2k2|edξ2+η2k21+αedξ2+η2k2(^gδˆg)|2dξdη+ξ2+η2<k2|eidk2ξ2η21+αeidk2ξ2η2(^gδˆg)|2dξdη)1/2supξ2+η2k2|edξ2+η2k21+αedξ2+η2k2|ˆgδˆg+supξ2+η2<k2|eidk2ξ2η21+αeidk2ξ2η2|ˆgδˆgsupξ2+η2k2|edξ2+η2k21+αedξ2+η2k2|δ+supξ2+η2<k2|eidk2ξ2η21+αeidk2ξ2η2|δδα+δ1α.

    If α=(δE)r with 0<r<1, then we have

    vαδ(,,0)vα(,,0)δ1rEr+δ1δrEr=δ1rEr(1+o(1)),forδ0. (2.20)

    (II) (Convergence) Via Parseval's identity, we have

    vα(,,0)u(,,0)=ˆvα(,,0)ˆu(,,0)=eda(ξ,η)1+αeda(ξ,η)ˆg(ξ,η)eda(ξ,η)ˆg(ξ,η)=αeda(ξ,η)1+αeda(ξ,η)ˆu(ξ,η,0)=αeda(ξ,η)(1+ξ2+η2)p/21+αeda(ξ,η)(1+ξ2+η2)p/2ˆu(ξ,η,0)=(ξ2+η2k2|αedξ2+η2k2(1+ξ2+η2)p/21+αedξ2+η2k2(1+ξ2+η2)p/2ˆu(ξ,η,0)|2dξdη+ξ2+η2<k2|αeidk2ξ2η2(1+ξ2+η2)p/21+αeidk2ξ2η2(1+ξ2+η2)p/2ˆu(ξ,η,0)|2dξdη)1/2Esupξ2+η2k2|αedξ2+η2k2(1+ξ2+η2)p/21+αedξ2+η2k2|+Esupξ2+η2<k2|αeidk2ξ2η2(1+ξ2+η2)p/21+αeidk2ξ2η2|:=I1+I2.

    For I2, by (2.11) and (1+ξ2+η2)p/21, we obtain

    I2=Esupξ2+η2<k2|αeidk2ξ2η2(1+ξ2+η2)p/21+αeidk2ξ2η2|αE1α=δrE1r1δrEr. (2.21)

    Next we estimate the term I1=Esupξ2+η2k2|αedξ2+η2k2(1+ξ2+η2)p/21+αedξ2+η2k2|. Denote ζ:=ξ2+η2k2, therefore, we need to estimate the term

    |αedξ2+η2k2(1+ξ2+η2)p/21+αedξ2+η2k2|=αedζ(1+k2+ζ2)p/21+αedζ.

    It is divided into two cases for ζ.

    Case I. ζ1dln(1α). It yields αedζα. From

    αedζ(1+k2+ζ2)p/21+αedζαedζα=δr/2Er/2.

    Case II. ζ1dln(1α). Firstly, we have

    αedζ(1+k2+ζ2)p/21+αedζαedζ(1+k2+ζ2)p/2αedζζp.

    Now we need to estimate ζp under the condition ζ1dln(1α).

    If ζ1dln(1α), we have

    ζp(1dln(1α))p=(r2dlnEδ)p.

    Therefore, combining (I) Stability and (II) Convergence, and using the triangle inequality, we have

    vαδ(,,0)u(,,0)O((lnEδ)p),forδ0.

    Remark 2.1. In this paper, we only consider the reconstruction of χA(x,y), where the domain A is a two-dimensional cube-like shape and belongs to the Sobolev space Hs(R2) with s<1/2. Therefore, in this theorem when 0<p<1/2, the result is convergence and when p<0, the result is divergence.

    In this section, we give the comparison between the quasi-boundary method and Tikhonov method in the situation of free noise. Firstly, we concentrate on the Tikhonov regularization method [34]. The method consists of looking for the solution for the inverse problem, and minimizes the quadratic functional, i.e., finding a minimizer uδα,tik(x,y,z) such that

    uδα,tik(x,y,z)=argminu{F(z)u(,,z)gδ(,)2+αu(,,0)2}, (3.1)

    where α=δ2/E2 with δ and E given by (2.1) and (2.2), respectively, F(z):u(x,y,z)g(x,y) is the forward operator of the problem. By Parseval's identity and according to (1.12), the variational problem becomes

    argminu{e(dz)a(ξ,η)ˆu(ξ,η,z)^gδ(ξ,η)2+αeza(ξ,η)ˆu(ξ,η,z)2}. (3.2)

    Let ˆuδα,tik(ξ,η,z) be the solution of above problem, then it satisfies the Euler equation

    [e2(dz)a(ξ,η)+αe2za(ξ,η)]ˆuδα,tik(ξ,η,z)=e(dz)a(ξ,η)^gδ(ξ,η). (3.3)

    The Tikhonov regularization solution ˆuδα,tik(ξ,η,z) in frequency domain can be given:

    ˆuδα,tik(ξ,η,z)=e(dz)a(ξ,η)1+αe2da(ξ,η)^gδ(ξ,η). (3.4)

    The restoration of Tikhonov method is then obtained by inverse Fourier transforming ˆuδα,tik(ξ,η,z). Under the conditions (2.1) and (2.2), we can prove the following error estimate for Tikhonov method in the same way:

    uδα,tik(,,z)u(,,z)2δzdEdzd(1+o(1)),forδ0. (3.5)

    It is natural for us to compare the quasi-boundary regularization method with the Tikhonov regularization when ξ2+η2k2. Now denote the norm of the regularization error vαu2 associated with the quasi-boundary regularization as ε1. Denote the norm of the regularization error uα,tiku2 associated with the Tikhonov regularization as ε. Thus, we have

    ε1=vαu2=R2(αedξ2+η2k21+αedξ2+η2k2)2e2(dz)ξ2+η2k2|ˆg(ξ,η)|2dξdη.
    ε=uα,tiku2=R2(αe2dξ2+η2k21+αe2dξ2+η2k2)2e2(dz)ξ2+η2k2|ˆg(ξ,η)|2dξdη.

    Now, the difference in regularization errors for Tikhonov method and the quasi-boundary method is given by

    d:=εε1=R2[(αe2dξ2+η2k21+αe2dξ2+η2k2)2(αedξ2+η2k21+αedξ2+η2k2)2]e2(dz)ξ2+η2k2|ˆg(ξ,η)|2dξdη.

    Obviously, the function H(y):=y1+αy for a fixed α>0 is an increasing function with respect to y>0. Therefore, d>0.

    Then we have the following result.

    Conclusion 3.1. In the absence of noise in the data, the quasi-boundary regularization method is more accurate than Tikhonov method in the sense of L2-error when the same parameter α is used.

    Remark 3.1. From the above comparison, we can give a general regularization method with noisy data based on the quasi-boundary method

    ^uαγ,δ(ξ,η,z)=e(dz)a(ξ,η)1+αeγda(ξ,η)^gδ(ξ,η),with1γ2. (3.6)

    Obviously, for γ=1, it is the quasi-boundary method. For γ=2, it is the Tikhonov method. For 1<γ<2, we call it the quasi-boundary-Tikhonov method.

    In this section in numerics, we consider the numerical computation in both cases of evanescent waves and homogeneous waves. In numerical computation, we refer to the following formulas. The quasi-boundary regularization solution:

    ^(vαδ)(ξ,η,0)=edξ2+η2k21+αedξ2+η2k2^gδ(ξ,η),ifξ2+η2k2,^(vαδ)(ξ,η,0)=edik2ξ2η21+αedik2ξ2η2^gδ(ξ,η),ifξ2+η2<k2. (4.1)

    The classical Tikhonov solution:

    ˆuδα,tik(ξ,η,0)=edξ2+η2k21+αe2dξ2+η2k2^gδ(ξ,η),ifξ2+η2k2,ˆuδα,tik(ξ,η,0)=edik2ξ2η21+αe2dik2ξ2η2^gδ(ξ,η),ifξ2+η2<k2. (4.2)

    In this section, we describe an experiment to illustrate the ability of the quasi-boundary regularization method for solving the inverse problem. Although the inverse problem is formulated in an unbounded domain in the (x,y) plane, we are interesting in the domain (x,y)[0,1]×[0,1]. This is reasonable because the problem can be solved by periodic extension to (x,y) plane.

    Define the discrete L2 norm of f(x,y) by

    f=(n2nj,k=1f(xj,yk)2)12, (4.3)

    where n is the total number of sampled points. In this section, we take n=100.

    In order to measure the accuracy of numerical results, we define the discrete L2-norm error ef for the exact solution f between the approximate solution fa as follow:

    ef=(n2nj,k=1|f(xj,yk)fa(xj,yk)|2)12. (4.4)

    The noise was added to g(x,y) by setting δ(xj,tk)=σrjkmax{g(xj,yk)}), where σ denotes the noise level and rjk a random number drawn from a uniform distribution in the range [1,1]. In numerical experiment, we use (4.3) to compute δ=ggδ. In this section, we used the discrete L2-norm E of the exact solution f(x,y) and the δ to compute the signal-to-noise ratio E/δ. In computation, according to the theoretical result the regularization parameters of the quasi-boundary method and Tikhonov method are chosen as α=δ/E and α=δ2/E2 respectively when the noise is added if necessary.

    We shall illustrate the reconstruction methods (4.1) and (4.2) with different numerical examples. The Fourier formulas (4.1) and (4.2) are based on FFT algorithm.

    Example 1. We choose the aperture A in the shape of the letter 'L' for the test. The characteristic function as shown in Figure 1a on the the domain A is to be constructed.

    Figure 1.  (a) The original A, (b) The data g.

    Convergence of the proposed method in near field pattern: In this experiment where d=0.1 and the wave-number k=2π and E=0.35, first we investigate the effectiveness of the proposed method, i.e., the convergence of the quasi-boundary method. Figure 1b displays data g which is a very blurred picture in the diffraction field. In Figure 2a, b, the σ=0.01 is changed to σ=0.0001, we can see the reconstructed result become better.

    Figure 2.  Quasi-boundary method (near field pattern): (a) α=2102, (b) α=2104.

    Convergence of the proposed method in far field pattern: In this experiment where d=0.2 and the wave-number k=20π and E=0.35, we investigate the the convergence of the quasi-boundary method. In Figure 3a, b, the σ=0.01 is changed to σ=0.0001, we can see the reconstructed result become better, but the rate is very slow.

    Figure 3.  Quasi-boundary method (far field pattern): (a) α=2102, (b) α=2104.

    From near field pattern to far field pattern: In this experiment the wave-number k=20π, λ=0.1, E=0.35 and σ=0.01, we investigate the result when we change the d. In Figure 4a, b, the d=0.05 (near field) is changed to d=0.5 (far field), we can see the reconstructed result become worse.

    Figure 4.  Quasi-boundary method: (a) Near field pattern: α=2102, (b) Far field pattern: α=2102.

    Quasi-boundary regularization method vs Tikhonov method: In this experiment, we take d=0.6 and the wave-number k=1, σ=0.01. In this case, to illustrate the Conclusion 3.1, firstly we add no perturbation to the data g(x,y). For the sake of fairness, in the numerical tests, the regularization parameters α is taken as α=1108 in the methods. Figure 5 corresponds to the results by the Tikhonov method and quasi-boundary method. From Figure 5, obviously the quasi-boundary regularization method shows better approximation than Tikhonov method. A lot of experiments show the same result. Here, we don't list the plots.

    Figure 5.  (a) Tikhonov regularization with α=1108, (b) Quasi-boundary method with α=1108.

    Example 2. The aperture A consists of two independent squares. The characteristic function to be constructed and the data g is displayed in Figure 6. In this example, 1% relative error is added to the data g(x,y) at the sampled points in the simulation as the Example 1.

    Figure 6.  (a) The original A, (b) The data g.

    According to [1], the Fourier truncated method is formulated as the following:

    ^(wαδ)(ξ,η,0)=edξ2+η2k21+αedξ2+η2k2^gδ(ξ,η),ifkξ2+η2keff,^(wαδ)(ξ,η,0)=0,ifξ2+η2>keff, (4.5)

    where keff=k[1+1(kd)2ln2(E/δ)].

    From this formula, we can see that the truncated Fourier regularized solution is a bandlimited function in the frequency, and the solution in the time field is analytic function.

    Figure 7 corresponds to the results by the truncated Fourier method and quasi-boundary method, where the distance from receiving plane to the screen is d=0.1 with the wave-number k=5.5.

    Figure 7.  (a) Fourier regularization with keff=389, (b) Quasi-boundary method with α=5102.

    In this numerical test, according to the theoretical result the regularization parameters of the quasi-boundary method and truncated Fourier method are chosen as α=δ/E and keff. From Figure 7, the truncated Fourier method shows its instability.

    In this paper, we gave a linear regularization method for solving the the reconstruction problem of aperture in the plane from their diffraction patterns. In theory, we proved the convergence of quasi-boundary regularization method. Relative to the classical Tikhonov method, for non-smooth reconstructing object, the quasi-boundary regularization show better numerical results (it can recover the "corner" of the object better) because it has less smoothing effect.

    The second author would like to thank Prof. Peijun Li (Department of Mathematics, Purdue University)'s help.

    This research was partially supported by the Natural Science Foundation of China (No. 11661072), the Natural Science Foundation of Northwest Normal University, China (No. NWNU-LKQN-17-5).

    The authors declare no conflicts of interest.



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