APPMATHISRN Applied Mathematics2090-55722090-5564International Scholarly Research Network43546810.5402/2012/435468435468Research ArticleA Wavelet Method for the Cauchy Problem for the Helmholtz EquationDouFang-Fang1FuChu-Li2BellouquidA.MarinL.1School of Mathematical SciencesUniversity of Electronic Science and Technology of China Chengdu 610054Chinauestc.edu.cn2School of Mathematics and StatisticsLanzhou UniversityLanzhou 730000Chinalzu.edu.cn201210102011201216082011280920112012Copyright © 2012 Fang-Fang Dou and Chu-Li Fu.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

1. Introduction

The Helmholtz equation is often used to approximate model wave propagation in inhomogeneous media. The demand for reliable numerical solutions to such type of problems is frequently encountered in geophysical and optoelectronic applications [1, 2]. In geophysical applications, for example, wave propagation simulations are used for the development of acoustic imaging techniques for gaining knowledge about geophysical structures deep within the Earth’s subsurface . In optoelectronics, the determination of a radiation field surrounding a source of radiation (e.g., a light emitting diode) is also a frequently occurring problem . In many engineering problems, the boundary conditions are often incomplete, either in the form of underspecified and overspecified boundary conditions on different parts of the boundary or the solution is prescribed at some internal points in the domain. These so-called Cauchy problems are inverse problems, and it is well known that they are generally ill posed in the sense of Hadamard . However, the Cauchy problem suffers from the nonexistence and instability of the solution.

In this paper we consider the Cauchy problem for the Helmholtz equation in a “strip” 0<x<1 as follows:Δu(x,y)+k2u(x,y)=0,  x(0,1),yRn,n1,u(0,y)=g(y),yRn,ux(0,y)=0,yRn, where Δ=2/x2+i=1n2/yi2 is an n+1 dimensional Laplace operator. We want to determine the solution u(x,y) for 0<x1 from the data g(y). Due to the importance of its application, this problem has been studied by many researchers, for example, DeLillo et al. [6, 7], Jin and Zheng , Johansson and Martin , and Marin et al. .

Let 𝒮 be the Schwartz space over n, and let 𝒮 be its dual (the space of tempered distributions). Let f̂ denote the Fourier transform of function f(y)𝒮 defined byf̂(ξ)=1(2π)n/2  Rne-iξyf(y)dy,  ξ=(ξ1,,ξn),y=(y1,,yn), while the Fourier transform of a tempered distribution f𝒮 is defined by(f̂,ϕ)=(f,ϕ̂),  ϕS. In this paper, we will consider functions depending on the variables x[0,1], yn.

For s, the Sobolev space Hs(n) consists of all tempered distributions f(y)𝒮, for which f̂(ξ)(1+|ξ|2)s/2 is a function in L2(n). The norm on this space is given byfHs:=(Rn|f̂(ξ)|2(1+|ξ|2)sdξ)1/2.

We assume there exists a unique solution u(x,y) of problem (1.1), which satisfies the problem in the classical sense and g(·), u(x,·)L2(n). Applying the Fourier transform technique to problem (1.1) with respect to the variable y yields the following problem in the frequency space:ûxx(x,ξ)+(k2-|ξ|2)û(x,ξ)=0,x(0,1),ξRn,n1,û(0,ξ)=ĝ(ξ),ξRn,ûx(0,ξ)=0,ξRn. It is easy to obtain the solution of problem (1.5) (if exists) has the formû(x,ξ)=cosh(x|ξ|2-k2)ĝ(ξ), or equivalently, the solution of problem (1.1) has the representationu(x,y)=1(2π)n/2Rneiξycosh(x|ξ|2-k2)ĝ(ξ)dξ. Since cosh(x|ξ|2-k2) increases rapidly with exponential order as |ξ|, the Fourier transform of the exact data g(y) must decay rapidly. However, in practice, the data at x=0 is often obtained on the basis of reading of physical instrument which is denoted by gm. We assume that g(·) and gm(·) satisfyg()-gm()Hrδ. Since gm(·) belong to L2(n)Hr(n) for r0, r should not be positive. A small perturbation in the data g(y) may cause a dramatically large error in the solution u(x,y) for 0<x1. Hence problem (1.1) is severely ill posed and its numerical simulation is very difficult. It is obvious that the ill-posedness of the problem is caused by the perturbation of high frequencies.

By (1.6) we knowû(1,ξ)=cosh(|ξ|2-k2)ĝ(ξ). Since the convergence rates can only be given under a priori assumptions on the exact solution , we will formulate such an a priori assumption in terms of the exact solution at x=1 by consideringu(1,)HsE.

Meyer wavelets are special because, unlike most other wavelets, they have compact support in the frequency domain but not in the time domain (however, they decay very fast). The wavelet methods have been used to solve one-dimensional heat conduction problems [16, 17] and noncharacteristic Cauchy problem for parabolic equation in one-dimensional  and multidimensional  cases, and so forth. In this paper we propose a similar wavelet method as suggested in  to the problem (1.1).

The paper is organized as follows. In Section 2 we describe the Meyer wavelets and discuss the properties that make them useful for solving ill-posed problems. Some error estimates between the exact solution and its approximation as well as the choice of the regularization parameter are given in Section 3. Finally, in Section 4 numerical tests verify the efficiency and accuracy of the proposed method.

2. The Meyer Wavelets

In the present paper let Φ be Meyer’s orthonormal scaling function in n dimensions. This function is constructed from the one-dimensional scaling functions in the following way. Let ϕ(x) and ψ(x) be the Meyer scaling and wavelet function in one dimension defined by their Fourier transform in  which satisfysuppϕ̂=[-43π,43π],suppψ̂=[-83π,-23π][23,  83π]. It can be proved (cf. ) that the set of functionsψjk(x)=2j/2ψ(2jx-k),  j,kZ, is an orthonormal basis of L2(). Consequently, the MRA {Vj}j of Meyer is generated byVj={ϕjk,kZ}¯,ϕjk:=2j/2ϕ(2jx-k),j,kZ,supp(ϕ̂jk)={ξ;  |ξ|43π2j}.

For the construction of an n-dimensional MRA, we take tensor products of the spaces Vj (see [21, 22]). Then the scaling function Φ is given byΦ(x)=k=1nϕ(xk),xRn, and any basis function Ψ in WJ can be written in the formΨ(x)=2nJ/2ψ(2Jxi-ki)miθm(2Jxm-km),  xRn, where kn, and for any m{1,,n}, θm stands for ϕ or ψ. Hence we obtain from (2.1) thatsupp(Φ̂)=[-43π,43π]n,f̂(ξ)=0  for  ξ23π2J,fWJ,JN. The orthogonal projection on the space VJ is defined byPJf:=kZn(f,ΦJ,k)ΦJ,k, while QJf denotes the orthogonal projection of a function f on the wavelet space WJ with VJ+1=VJWJ. (In many contexts one will find more than one detailed space WJ, that is, VJ+1=VJW1,JW2,J. Here, the space WJ is simply defined as the orthogonal complement of VJ in VJ+1).

LetΩJ:=2J[-23π,23π]n. Setting ΓJ:=nΩJ, together with (2.6), it follows for J thatPJf̂(ξ)=0forξΓJ+1,((I-PJ)f)̂(ξ)=QJf̂(ξ)forξΩJ+1. We introduce the operator MJ which is defined by the equationMJf̂:=(1-χJ)f̂,  JN, where χJ denotes the characteristic function of the cube ΩJ. From (2.7) it follows that any basis function Ψ in Wj, jJ, satisfiesΨ̂(ξ)=0,  ξΩJ, and we obtain(f,Ψ)=(f̂,Ψ̂)=((1-χJ)f̂,Ψ̂)=(MJf,Ψ). And it follows for J thatQJ=QJMJ,I-PJ=(I-PJ)MJ.

3. Wavelet Regularization and Error Estimates

We list the following two lemmas given in [19, 23] which are useful to our proof.

Lemma 3.1 (see [<xref ref-type="bibr" rid="B12">19</xref>, <xref ref-type="bibr" rid="B25">23</xref>]).

Let {VJ}J be an m-regular MRA, and let r,s be such that -m<r<s<m. Then for each function fHs(n) and J, the following inequality holds: f-PJfHrC12-J(s-r)fHr.

Lemma 3.2 (see [<xref ref-type="bibr" rid="B12">19</xref>]).

Let {VJ}J be Meyer’s (tensor-) MRA, and suppose J, r. Then for all fVJ, one has lxilfHrC22(J-1)lfHr,  i=1,,n,lN.

Define an operator Tx:g(y)u(x,y) by (1.6), that is,Txg=u(x,y),  0<x1, or equivalently,Txĝ(ξ)=cosh(x|ξ|2-k2)ĝ(ξ),0<x1. Then we have

Theorem 3.3.

Let {VJ}J be Meyer’s MRA and suppose r and J which satisfies 2J>k, 0x1. Then for all fVJ, one has TxfHr(C5ex22(J-1)-k2+1)fHr.

Proof.

For fVJ, by definition (1.4) and formula (3.4), from Lemma 3.2, we have TxfHr=(Rn|cosh(x|ξ|2-k2)f̂|2(1+|ξ|2)rdξ)1/2(|ξ|>k|cosh(x|ξ|2-k2)cosh(x|ξ|)cosh(x|ξ|)f̂|2(1+|ξ|2)rdξ)1/2+(|ξ|k|cos(xk2-|ξ|2)f̂|2(1+|ξ|2)rdξ)1/2supξΩJ+1|cosh(x|ξ|2-k2)cosh(x|ξ|)|(|ξ|>k|cosh(x|ξ|)f̂|2(1+|ξ|2)rdξ)1/2+fHr2supξΩJ+1ex(|ξ|2-k2-|ξ|)(|ξ|>k|l=0x2l(2l)!|ξ|2lf̂|2(1+|ξ|2)rdξ)1/2+fHr2C3ex(22(J-1)-k2-2J-1)l=0x2l(2l)!(Δy)lfHr+fHr(C4ex(22(J-1)-k2-2J-1)l=0x2l(2l)!n22(J-1)l+1)fHr(C5ex(22(J-1)-k2-2J-1)cosh(x2J-1)+1)fHr(C5ex22(J-1)-k2+1)fHr.

Since the Cauchy data are given inexactly by gm, we need a stable algorithm to approximate the solution of (1.1). Our method is as follows. Consider the operatorTx,J:=PJTxPJ, and show that it approximates Tx in a stable way for an appropriate choice for J depending on δ and E. By the triangle inequality we knowTxg-Tx,JgmHr(Tx-Tx,J)gHr+Tx,J(g-gm)Hr.

From (1.8) and Theorem 3.3, the second term on the right-hand side of (3.8) satisfiesTx,J(g-gm)Hr=PJTxPJ(g-gm)HrTxPJ(g-gm)Hr(C5ex22(J-1)-k2+1)δ.

For the first one we have(Tx-Tx,J)gHr(I-PJ)TxgHr+PJTx(I-PJ)gHr.

By Lemma 3.1, (1.9), (1.10), (2.14), and (3.4), we get(I-PJ)TxgHr=(I-PJ)MJTxgHrC12-J(s-r)MJTxgHs=C12-J(s-r)(ΓJ|cosh(x|ξ|2-k2)cosh(|ξ|2-k2)u(1,)|2(1+|ξ|2)s)1/2C12-J(s-r)supξΓJ|cosh(x|ξ|2-k2)cosh(|ξ|2-k2)|u(1,)Hs2C1((32π2J)2-k2)-(s-r)/2e-(1-x)((2/3)π2J)2-k2E2C1(22J-k2)-(s-r)/2e-(1-x)22J-k2E. On the other hand, due to (2.10), we knowPJTx(I-PJ)gHrTx(I-PJ)gHr(ΩJ+1|cosh(x|ξ|2-k2)(QJg)̂(ξ)|2(1+|ξ|2)rdξ)1/2+(ΓJ+1|cosh(x|ξ|2-k2)ĝ(ξ)|2(1+|ξ|2)rdξ)1/2=:I1+I2. We estimate the two parts at the right-hand side of (3.12) separately. For I2 we haveI2=(ΓJ+1|cosh(x|ξ|2-k2)ĝ(ξ)|2(1+|ξ|2)rdξ)1/2=(ΓJ+1|cosh(x|ξ|2-k2  )cosh(|ξ|2-k2)u(1,)|2(1+|ξ|2)rdξ)1/2supξΓJ+1|cosh(x|ξ|2-k2)cosh(|ξ|2-k2)1(1+|ξ|2)(s-r)/2|×(ΓJ+1|u(1,)(1+|ξ|2)s/2|2dξ)1/22(43π2J)-(s-r)e-(1-x)n((4/3)π2J)2-k2u(1,)Hs2(22J-k2)-(s-r)/2e-(1-x)22J-k2E. Now we turn to I1. There holdsI1=(ΩJ+1|cosh(x|ξ|2-k2)(QJg)̂(ξ)|2(1+|ξ|2)rdξ)1/2TxQJgHr(C5ex22J-k2+1)QJgHr, since QJgVJ+1. Furthermore, from (2.14), it follows thatQJgHr=QJMJgHrMJgHr=(ΓJ|ĝ(ξ)|2(1+|ξ|2)rdξ)1/2=(ΓJ|û(1,ξ)cosh(|ξ|2-k2)|2(1+|ξ|2)rdξ)1/222-J(s-r)e-n((2/3)π2J)2-k2u(1,)Hs2(22J-k2)-(s-r)/2e-22J-k2E. Therefore,PJTx(I-PJ)gHr2(1+C5)(22J-k2)-(s-r)/2e-(1-x)22J-k2E+2(22J-k2)-(s-r)/2e-22J-k2E. Combining (3.11) and (3.16) with (3.10), we have(Tx-Tx,J)gHr2(C1+1+C5)(22J-k2)-(s-r)/2e-(1-x)22J-k2E+2(22J-k2)-(s-r)/2e-22J-k2E. Then from (3.9) and (3.17) we finally arrive atTxg-Tx,JgmHr(C5ex22(J-1)-k2+1)δ+2(22J-k2)-(s-r)/2e-22J-k2E+2(C1+1+C5)(22J-k2)-(s-r)/2e-(1-x)22J-k2E.

In order to show some stability estimates of the Hölder type for our method using (3.18), we use the following lemma which appeared in  for choosing a proper regularization parameter J.

Lemma 3.4.

Let the function f(λ):[0,a] be given by f(λ)=λb(dln1λ)-c with a constant c and positive constants a<1, b, and d. Then for the inverse function f-1(λ), one has f-1(λ)=λ1/b(dbln1λ)c/b(1+o(1))forλ0.

Based on this lemma, we can choose the regularization parameter J by minimizing the right-hand side of (3.18).

Denotee-22J-k2=λ(0,1), and let C=C5/2(1+C1+C5) andCλ-xδ=λ1-x(ln1λ)-(s-r)E, that is,CδE=λ(ln1λ)-(s-r). Then by Lemma 3.4 we obtain thatλ=CδE(ln1Cδ/E)s-r(1+o(1))forCδE0=CδE(lnECδ)s-r(1+o(1))forδ0. Taking the principal part of λ, we getJ*=12log2(ln2(ECδ(lnECδ)-(s-r))+k2), due to (3.21). Now, summarizing above inference process, we obtain the main result of the present paper.

Theorem 3.5.

For sr, suppose that conditions (1.8) and (1.10) hold. If one takes J**=[J*], where J* was defined in (3.25), [a] with square bracket denotes the largest integer less than or equal to aR. Then there holds the following stability estimate: Txg-Tx,Jgm(2(C1+1+C5)E)x(C5δ)1-x(lnECδ)-(s-r)x×(1+(lnE/CδlnE/Cδ+ln(lnE/Cδ)-(s-r))s-r)+(1+2C(lnE/CδlnE/Cδ+ln(lnE/Cδ)-(s-r))s-r)δ=(2(C1+1+C5)E)x(C5δ)1-x(lnECδ)-(s-r)x(1+o(1)), for δ0.

Remark 3.6.

In general, the a priori bound E and the coefficients C1-C5 and C are not exactly known in practice. In this case, with J=[12log2(ln2(1δ(ln1δ)-(s-r))+k2)], it holds that Txg-Tx,Jgmδ1-x(ln1δ)-(s-r)x(C5+2(C1+1+C5)E(ln1/δln1/δ+ln(ln1/δ)-(s-r))s-r)+(C5+2E(ln1/δln1/δ+ln(ln1/δ)-(s-r))s-r)δ=δ1-x(ln1δ)-(s-r)x(1+o(1)), for δ0.

Remark 3.7.

The proposed wavelet method can also be used to solve the following Cauchy problem for the modified Helmholtz equation (i.e., the Yukawa equation ) Δv(x,y)+k2v(x,y)=0,x(0,1),yRn,n1,v(0,y)=g(y),yRn,vx(0,y)=0,yRn, where Δ=2/x2+i=1n2/Δyi2 is the same as in (1.1).

It is easy to know that the exact solution of problem (3.30) is v(x,y)=1(2π)n/2Rneiξycosh(x|ξ|2+k2)ĝ(ξ)dξ. Define an operator T̃x:g(y)v(x,y) such that T̃xĝ(ξ)=cosh(x|ξ|2+k2)ĝ(ξ),0<x1, and the approximate solution is vJδ=T̃x,Jgm, where g and gm satisfy (1.8), and T̃x,J=PJT̃xPJ. If we select the regularization parameter J=[12log2(ln2(1δ(ln1δ)-(s-r))-k2)], then there holds v(x,)-vJδ(x,)δ1-x(ln1δ)-(s-r)x×(C5+2(C1+1+C5)E(ln1/δln1/δ+ln(ln1/δ)-(s-r))s-r)+(C5+2E(ln1/δln1/δ+ln(ln1/δ)-(s-r))s-r)δ=δ1-x(ln1δ)-(s-r)x(1+(o(1))), for δ0.

4. Numerical Aspect4.1. Numerical Implementation

We want to discuss some numerical aspects of the proposed method in this section.

We consider the case when n=2. Supposing that the sequence {g(y1,i,y2,j)}i,j=1N represents samples from the function g(y1,y2) on an equidistant grid in the square [a,b]2, and N is even, then we add a random uniformly distributed perturbation to each data and obtain the perturbation datagm=g+μrandn(size(g)). Then the total noise δ can be measured in the sense of root mean square error according toδ:=gm-gl2=1N2i=1Nj=1N(gm(y1,i,y2,j)-g(y1,i,y2,j))2, where randn(·)" is a normally distributed random variable with zero mean and unit standard deviation and ϵ dictates the level of noise. randn(size(g))" returns an array of random entries that is the same size as g.

For the function gm(y1,y2), we haveuJδ(x,y)=Tx,Jgm=PJTxPJgm. Hence, by using it with J being given in (3.28), we can obtain the approximate solution.

We will use DMT as a short form of the “discrete Meyer (wavelet) transform." Algorithms for discretely implementing the Meyer wavelet transform are described in . These algorithms are based on the fast Fourier transform (FFT), and computing the DMT of a vector in requires 𝒪(Nlog22N) operations.

4.2. Numerical Tests

In this section some numerical tests are presented to demonstrate the usefulness of the approach. The tests were performed using Matlab and the wavelet package WaveLab 850, which was downloaded from http://www-stat.stanford.edu/~wavelab/. Throughout this section, we set μ=10-3, a=-5, b=5, and N=26.

Example 4.1.

Take n=2 and g(y)=e-y2𝒮(2), where y=(y1,y2) and 𝒮(2) denotes the Schwartz function space.

Since ĝ(ξ)𝒮(2), ξ=(ξ1,ξ2) decays rapidly, and the formula (1.7) can be used to calculate u(x,y) with exact data directly, that is, u(x,y)=12πR2ei(ξ1y1+ξ2y2)cosh(xξ12+ξ22-k2)ĝ(ξ1,ξ2)dξ1dξ2.

In Figure 1 we give the exact solution at x=1, that is, u(1,y1,y2), and the reconstructed solution uδ(1,y1,y2) from the noisy data gm(y1,y2) without regularization. We see that uδ does not approximate the solution and some regularization procedure is necessary.

(a) Exact solution u(1,·); (b) unregularized solution reconstructed from gm for x=1.

Letting k=1, the regularized solutions and the corresponding errors u-uJδ defined by the regularization parameter J=3,4,5 are illustrated in Figure 2. We can see that in V3 the approximation is very poor since the frequencies are cut off excessively by the projection P3. If J is taken to be too large, the noise in the function gm is not damped enough by PJ, and thus the high frequencies of ĝm are so extremely magnified that they destroy the approximated solution. The approximation parameter J=4 seems to be the optimal choice for this example.

(a) The regularized solution at x=1; (b) difference between the regularization and the exact solution; (1), (2), (3) correspond to J=3,4,5, respectively.

In Figure 3 we display the exact solution, its approximation, and corresponding errors for k=5 and 100, respectively. We see that the proposed method is useful for different wave number k.

(a) and (b) correspond to k=5 and k=100, respectively; (1), (2), (3) correspond to the exact solution, the regularized solution and the difference between the regularization and the exact solution, respectively.

Figure 4 shows that the proposed method for the Cauchy problem for the modified Helmholtz equation is also effective.

(a) Exact solution v(1,·); (b) unregularized solution reconstructed from gm for x=1; (c) regularized solution reconstructed from gm for x=1 and J=4; (d) the difference between the regularization and the exact solution.

Acknowledgments

The authors would like to thank the WaveLab Team at the Stanford University for the help of their wavelet package Wavelab850. The work described in this paper was supported by the Fundamental Research Funds for the Central Universities of China (Project no. ZYGX2009J099) and the National Natural Science Foundation of China (Project no. 11171136).

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