A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data

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Applied Mathematical Modelling xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data q Chu-Li Fu a,⇑, Yun-Jie Ma b, Yuan-Xiang Zhang a, Fan Yang a,c a

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China School of Mathematics and Informational Science, Yantai University, Yantai 264005, PR China c School of Science, Lanzhou University of Technology, Lanzhou 730050, PR China b

a r t i c l e

i n f o

Article history: Received 3 September 2013 Accepted 15 December 2014 Available online xxxx Keywords: Ill-posed problem Cauchy problem for the Helmholtz equation A posteriori regularization Neumann data

a b s t r a c t In this paper the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data is considered. This problem is severely ill-posed, the solution does not depend continuously on the data. An approximate method based on the a posteriori Fourier regularization in the frequency space is analyzed. Some crucial information about the regularization parameter hidden in the a posteriori choice rule are found, and some sharp error estimates between the exact solution and its regularization approximate solution are proved. Numerical examples show the effectiveness of the method. A comparison of numerical effect between the a posteriori and the a priori Fourier method is also taken into account. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In this paper we speciﬁcally consider a Cauchy problem for the Helmholtz equation with only inhomogeneous Neumann data. The Helmholtz equation arises naturally in many physical applications, in particular related to acoustic or electromagnetic wave propagation. The Cauchy problem for the Helmholtz equation with a real wave number k on a ‘‘strip’’ domain is as follows:

8 2 n > < Duðx; yÞ þ k uðx; yÞ ¼ 0; x 2 ð0; 1Þ; y 2 R ; n uð0; yÞ ¼ uðyÞ; y2R ; > : y 2 Rn ; ux ð0; yÞ ¼ wðyÞ;

ð1:1Þ

Pn @ 2 2 n @2 where D ¼ @x 2 þ j¼1 @y2 is a ðn þ 1Þ-dimensional Laplace operator, uðyÞ; wðyÞ 2 L ðR Þ are the Dirichlet and Neumann data, j respectively. Some reasons for investigating this problem following from optoelectronics and in particular in laser beam models have been explained in detail in [1]. The Cauchy problem for the Helmholtz equation is an inverse problem [2] and is severely ill-posed, i.e., the solution does not depend continuously on the data, any small change of the data may cause dramatically large error in the solution. Therefore, some regularization methods for solving this problem is important and necessary. Due to the linearity of problem (1.1), it can be divided into two problems with only one inhomogeneous Dirichlet or Neumann data, respectively: q The project is supported by the NNSF of China (No. 11171136), the Tianyuan Fund for Mathematics of the NSF of China (No. 11326235), the Fundamental Research Funds for the Central Universities (Nos. lzujbky-2014-19, lzujbky-2014-24). ⇑ Corresponding author.

http://dx.doi.org/10.1016/j.apm.2014.12.030 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.

Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

8 2 n > < Dv ðx; yÞ þ k v ðx; yÞ ¼ 0; x 2 ð0; 1Þ; y 2 R ; n v ð0; yÞ ¼ uðyÞ; y2R ; > : v x ð0; yÞ ¼ 0; y 2 Rn

ð1:2Þ

8 2 n > < Dwðx; yÞ þ k wðx; yÞ ¼ 0; x 2 ð0; 1Þ; y 2 R ; n wð0; yÞ ¼ 0; y2R ; > : wx ð0; yÞ ¼ wðyÞ; y 2 Rn :

ð1:3Þ

and

It is obvious that if v ðx; yÞ and wðx; yÞ are the solutions of problem (1.2) and (1.3), respectively, then uðx; yÞ ¼ v ðx; yÞ þ wðx; yÞ is the solution of problem (1.1). For problem (1.2), there have been many results and various methods have also been presented. A a priori Fourier regularization method was ﬁrstly applied in [1], subsequently, this method has been further improved in [3]. Under the L2 ðRn Þ a priori bound assumption, the optimal error bound for approximate solution was derived and also the spectral and a revised Tikhonov regularization methods were presented in [4]. The conditional stability estimates in the general source condition was used to the optimality analysis of the approximate solution in [5]. Two quasi-reversibility methods were presented in [6,7], respectively. A modiﬁed Tikhonov method was presented in [8]. A quasi-boundary-value method was presented in [9]. As an application of the general theory of the a priori Fourier method, Example 3.3 in [10] also considered this problem. The regularization methods used in the vast majority of works on problem (1.2) are a priori methods, only [5,11] involve a posteriori regularization methods. However, to the authors’ knowledge, there are few works devoted to the error estimates of regularization methods for problem (1.3), in which the a priori Fourier method was applied in [10], and both the a priori and the a posteriori modiﬁed Tikhonov method was applied in [12]. In this paper we will apply the a posteriori Fourier method to solve problem (1.3). The reasons are listed below: 1. For any linear ill-posed problems deﬁned on a ‘‘strip’’ domain, the Fourier method, which was ﬁrst applied to the inverse heat conduction problem by Eldén et al. in [13], is the most simple and a very effective regularization method. 2. The general theory under the frame of numerical pseudodifferential operators for the a priori and the a posteriori Fourier method was preliminarily established in [10,11], respectively. It’s a little pity that the a posteriori theory given in [11] is unable to cover the Cauchy problem for the Laplace equation with inhomogeneous Neumann data and more is not applicable to problem (1.3). For the former, it has been considered in [14], but relatively, the research of problem (1.3) has more particular difﬁculty and also has independent signiﬁcance for perfecting the theory of the a posteriori Fourier method. 3. For any the a priori regularization method, the choice of the regularization parameter usually depends on both the a priori bound and the noise level. In general, the a priori bound cannot be known exactly in practice, and working with a wrong a priori bound may lead to bad regularization solution. The advantage of the a posteriori method is that one does not need to know the smoothness and the a priori bound of unknown solution. So it is particularly worthy of further development. However, because some important information about the solution are concealed and hidden for the Morozov’s discrepancy principle, such that the theoretical analysis of the convergence rate with high-accuracy of the approximate solution is rather difﬁcult. For example, the existing works for problem (1.3) about the a posteriori method only involve weaker L2 -smoothness assumption [5,12]. Therefore, some new ideas are needed to obtain more profound and deeper results for problem (1.3). 4. For problem (1.1)–(1.3), some rather sharp restriction for the wave number k was imposed in some known important works. For example, the condition dk < p2 was required in [1,5], where d is the width of the ‘‘strip’’ domain (without loss of generality, we take d ¼ 1 in this paper, i.e., x 2 ð0; 1Þ). Such a restriction will bring inconvenience to practical application, and some improvements are needed. The main aim of this paper is to solve the above puzzles effectively using the a posteriori Fourier method for problem (1.3). This paper is organized as follows. In Section 2, a new non-standard a priori bound for unknown solution is introduced, which is a reﬁnement of the standard Hp ðRn Þ-a priori bound and can more accurately portray the degree of illposedness for problem (1.3) in the exponential sense [15,16,10,11]. Meanwhile, a conditional stability result is also given, which will ensure the theoretical stringency of the a posteriori rule in the next section. In Section 3, an a posteriori choice rule of regularization parameter based on the Morozov’s discrepancy principle is given, and some deep-seated nature of the regularization parameter is stated, but the proof of main result is given in the appendix at the end of this paper. In Section 4, some convergence rate estimates with Hölder type or asymptotic logarithm-Hölder type are given. In Section 5, some numerical examples are provided, which show the effectiveness of the method. A comparison of numerical effect between the a posteriori and the a priori Fourier method is also taken into account in Example 5.1. This paper ends with a brief conclusion in Section 6. Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

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2. A new a priori bound and a conditional stability result The solution of problem (1.3) is given by [3]

Z

1 wðx; yÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ n ð 2p Þ

iyn

e

Rn

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 sinh x jnj2 k ^ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞdn 2 jnj2 k

ð2:1Þ

or equivalently,

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 sinh x jnj2 k ^ ^ nÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ; wðx; 2 2 jnj k

ð2:2Þ

^ nÞ denotes the Fourier transform of function wðx; yÞ with respect to variable y 2 Rn . where wðx; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 Noting that the factor pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ in (2.1) and (2.2) increase exponentially as jnj ! 1, so the problem is severely ill-posed. 2 2 jnj k

In practice the Neumann data wðyÞ is given only by measurement. Assume that the exact data wðyÞ and the noisy data wd ðyÞ both belong to L2 ðRn Þ and satisfy the following noise level

kwðÞ wd ðÞk 6 d;

ð2:3Þ 2

n

where k k denotes the norm in L ðR Þ. When w in (2.1) is replaced by wd , the error given by (2.3) must be ampliﬁed inﬁnitely pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jnj2 k2 by factor sinhpxﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ which leads to the blow-up of integral (2.1). So, the integral (2.1) with noisy data is a computational probjnj2 k2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jnj2 k2 lem of numerical pseudodifferential operator with the symbol sinhpxﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ [10], and some regularization method is needed. 2 2 jnj k

The standard a priori bound for problem (1.3) is given by [1,3,5]

kwð1; Þkp 6 E;

p P 0;

ð2:4Þ

which is deﬁned by

kwð1; Þkp :¼

Z

2 p

^ jwð1; Þj2 ð1 þ jnj Þ dn

12

Rn

0 Z B ¼@

2 112 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh jnj2 k2 p ^ ð1 þ jnj2 Þ dnC qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ A: 2 2 Rn jnj k

ð2:5Þ

When p ¼ 0; kwð1; Þkp ¼ kwð1; Þk just is the most commonly used L2 -a priori bound [1,11,12]. The main subject of this paper is to apply the a posteriori Fourier regularization method for problem (1.3), however, it is too hard to obtain the error estimates and convergence rate analysis of approximate solution by using the a priori bound (2.4). In order to overcome this difﬁculty, we will use the following new a priori bound:

Z jnjPk

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ^ 2 p wðnÞe jnj k ð1 þ jnj Þ dn

!12

6 E;

p P 0:

ð2:6Þ

Note that for jnj P k; 0 < x < 1,

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2nþ1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2n qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 2nþ1 2 2 jnj jnj x k x k x jnj2 k 1 1 1 X X X sinh x jnj k 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 6 ð2n þ 1Þ! ð2n þ 1Þ! ð2nÞ! 2 2 n¼0 n¼0 jnj2 k jnj2 k n¼0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ¼ cosh x jnj2 k 6 ex jnj k

ð2:7Þ

and for jnj 6 k; x > 0, there holds

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 sin x k2 jnj2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 6 x; 2 2 jnj2 k k jnj2

ð2:8Þ

we know that if (2.6) holds, then

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh jnj2 k2 2 2 p ^ kwð1; Þkp ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ ð1 þ jnj Þ dn n 2 R jnj2 k 2 2 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z sinh jnj2 k2 sinh jnj2 k2 p p 2 ^ ^ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q q ¼ wðnÞ ð1 þ jnj Þ dn þ wðnÞ ð1 þ jnj2 Þ dn 2 2 2 2 jnj6k jnjPk jnj k jnj k Z Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 p 2 p p 2 jnj k ^ 2 2 ^ ^ jwðnÞj ð1 þ jnj2 Þ dn þ jwðnÞj ð1 þ jnj2 Þ dn þ E2 : wðnÞ ð1 þ jnj2 Þ dn 6 6 e Z

jnj6k

jnjPk

jnj6k

Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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According to wðyÞ 2 L2 ðRn Þ, we know that the integral in the right hand side of the above inequality is convergent, and therefore the a priori bound (2.4) must be hold for another constant E. Conversely, assume that the standard the a priori bound (2.4) holds for order p þ 1, i.e.,

kwð1; Þk2pþ1

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh jnj2 k2 2 pþ1 2 ^ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ ð1 þ jnj Þ dn 6 E ; n 2 2 R jnj k Z

p P 0:

ð2:9Þ

Note that for jnj P k, there holds

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh jnj2 k2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2^ 2 pþ1 ^ ð1 þ jnj2 Þp ^ ð1 þ jnj2 Þp ¼ 1 e jnj2 k2 e jnj2 k2 wðnÞ P sinh jnj2 k wðnÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ ð1 þ jnj Þ 4 2 jnj2 k 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ^ ^ 2jwðnÞj ^ ð1 þ jnj2 Þp ; ¼ þ e jnj k wðnÞ e jnj k wðnÞ 4 therefore,

Z jnjPk

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z sinh jnj2 k2 p 2 pþ1 2 ^ ^ jwðnÞj ð1 þ jnj2 Þ dn: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ ð1 þ jnj Þ dn þ 2 2 2 jnjPk jnjPk jnj k

Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ dn 6 4 e

ð2:10Þ

It is obviously that if (2.9) holds, then both integrals on the right hand side of inequality (2.10) must be convergent, and therefore the a priori bound (2.6) holds for another constant E. Conclusion 2.1. If the a priori bound (2.6) holds, then the standard a priori bound (2.4) also holds with the same order p. Conversely, if the standard a priori bound (2.4) holds with order p þ 1, then the a priori bound (2.6) holds with order p. Remark 2.1. The new a priori bound (2.6) reﬂects the essential characteristics of the severe ill-posedness for problem (1.3). Meanwhile, there is a close relationship with the standard a priori bound. The new a priori bound (2.6) is a reﬁnement of the standard bound (2.4) between order p and p þ 1. Now we give a conditional stability result under the a priori bound (2.6), which is useful in the next section. Theorem 2.1. Suppose that the a priori bound (2.6) hold and wðx; yÞ is the solution of problem (1.3) given by formula (2.1) with the exact data wðyÞ. Then there holds the following estimate

kwðx; Þk2 6 x2 kwk2 þ E2x kwk2ð1xÞ ; 2

0 < x < 1;

ð2:11Þ

n

where k k denotes the L ðR Þ norm. Proof. Due to the Parseval formula, expressions (2.1), (2.2), inequalities (2.7), (2.8), condition (2.6), and the Hölder inequality, we have

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 2 ^ kwðx; Þk ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn n 2 2 n2R jnj k 2 2 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z sinh x jnj2 k2 sinh x jnj2 k2 ^ ^ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn 2 2 2 2 jnj6k jnjPk jnj k jnj k Z Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x jnj2 k2 ^ 2 ^ wðnÞ dn jwðnÞj dn þ 6 x2 e Z

n2Rn 2

¼ x2 kwk þ

jnjPk

Z jnjPk

6 x2 kwk2 þ

Z

jnjPk 2

2

6 x kwk þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ px px jnj2 k2 ^ 2x 2ð1xÞ ^ wðnÞ ð1 þ jnj2 Þ ð1 þ jnj2 Þ jwðnÞj dn e pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x 1x p jnj2 k2 ^ 2 2 ^ wðnÞ ð1 þ jnj2 Þ jwðnÞj dn e

Z jnjPk

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ dn e

!x Z

!1x 2 ^ jwðnÞj dn

jnjPk

6 x2 kwk2 þ E2x kwk2ð1xÞ :

Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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The estimate (2.11) is proved. h The conclusion given by Theorem 2.1 shows that if w1 ðx; yÞ and w2 ðx; yÞ are the solutions of problem (1.3) with the exact data w1 ðyÞ and w2 ðyÞ, respectively, then there holds

12 kw1 ðx; Þ w2 ðx; Þk 6 x2 kw1 w2 k2 þ E2x kw1 w2 k2ð1xÞ : In other words, if kw1 w2 k ! 0, then kw1 ðx; Þ w2 ðx; Þk ! 0 for 0 6 x < 1. 3. The a posteriori parameter choice rule for Fourier method and some auxiliary results Generally speaking, the conditional stability result cannot ensure the stability of numerical computation with noisy data for ill-posed problems. Therefore some effective regularization methods for solving problem (1.3) are important and necessary. The Fourier method is a rather simple and very effective regularization method for solving many ill-posed problems, such as inverse heat conduction problem [13], backward heat conduction problem [17], the Cauchy problem for the Laplace equation [18], numerical analytic continuation [19], the identiﬁcation of unknown source [20], etc. However, most of the known works with this method are limited to the a priori choice of the regularization parameter. Although the recent work [11] has systematically considered the general theory of the a posteriori Fourier method, it cannot apply to the theoretical analysis of the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, because the a priori information used is standard. In addition, [11] only consider the weaker L2 -a priori bound and the Hölder type’s convergence. Below we will further consider solving problem (1.3) with the a posteriori Fourier method and provide some ﬁner results. As in [13,3,10], deﬁne the function vnmax by

vnmax ¼

1; jnj 6 nmax ; 0;

ð3:1Þ

jnj > nmax ;

where nmax is a constant to be determined. Deﬁne the Fourier regularization approximate solution of problem (1.3) with noisy data wd ðyÞ 2 L2 ðRn Þ as

1 wd;nmax ðx; yÞ :¼ pﬃﬃﬃﬃﬃﬃﬃ n ð 2pÞ

Z

iyn

e

Rn

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 sinh x jnj2 k ^ d ðnÞv dn; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ w nmax 2 jnj2 k

ð3:2Þ

where nmax is regarded as the regularization parameter. Due to the Morozov’s discrepancy principle [21], we will take nmax as the solution of equation

^ d ðnÞk ¼ sd; kð1 vnmax Þw

ð3:3Þ

where s > 1 is a constant. ^ d k is continuous, decreasing and As in [22], it is easy to see that nmax # kð1 vnmax Þw

^ d k ¼ 0; lim kð1 vnmax Þw

nmax !þ1

^ d k ¼ kw ^ d k ¼ kw k: lim kð1 vnmax Þw d

nmax !0

In practice, there always holds kwd k > d, otherwise, wd;nmax 0 would be an acceptable approximation to wðx; yÞ. In fact, wd;nmax ðx; yÞ 0 can also be seen an exact solution of problem (1.3) with the exact data wðyÞ 0. Therefore, according to Theorem 2.1 we know, for 0 6 x < 1, there holds

kwd;nmax ðx; Þ wðx; Þk2 ¼ k0 wðx; Þk2 ¼ kwðx; Þk2 6 x2 kwk2 þ E2x kwk2ð1xÞ ¼ x2 kw wd þ wd k2 þ E2x kw wd þ wd k2ð1xÞ 6 x2 ðkw wd k þ kwd kÞ2 þ E2x ðkw wd k þ kwd kÞ2ð1xÞ 6 x2 ð2dÞ2 þ E2x ð2dÞ2ð1xÞ ! 0;

for d ! 0:

Therefore, for an appropriate constant s > 1, Eq. (3.3) is always solvable, and if the solution of Eq. (3.3) is not unique, then nmax will be understood as the minimal solution of this equation. It is easy to see from Eq. (3.3) that the choice of the regularization parameter nmax is only related to the noise level d but not dependent on the a priori bound. This is also the advantage of the a posteriori method. However, some important information has been hidden here, for example, how to judge the inﬂuence of the a priori bound on the value range of regularization parameter? These hidden information are crucial for rigorous theoretical analysis for convergent rate of approximate solution. The following theorem answers these particular concerns. Since the proof is quite technical and involved, we only state the results here and transfer the proof to the appendix at the end of the paper.

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Theorem 3.1. Assume that the conditions (2.3), (2.6) hold, the regularization parameter nmax is taken as the solution of Eq. (3.3). If nmax > k, then there hold

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 2

E for p ¼ 0: ðs 1Þd p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 2 E E ln ð1 þ oð1ÞÞ for d ! 0; p > 0: ðiiÞ e nmax k 6 ðs 1Þd ðs 1Þd

ðiÞ e

nmax k

ð3:4Þ

6

ð3:5Þ

ðiiiÞ For jnj P nmax ; there is a constant c > 1; such that p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp E 2 6 ð1 þ oð1ÞÞ ln for d ! 0; p > 0: jnj2 k ðs 1Þcd

ð3:6Þ

4. The error estimates for the a posteriori Fourier method In this section we will give some error estimates between the regularization approximate solution given by (3.2) with the a posteriori choice of nmax and the exact solution of problem (1.3) given by (2.1). Theorem 4.1. Assume the conditions (2.3) and (2.6) with p > 0 hold, if the regularization parameter nmax is chosen as the solution of Eq. (3.3), then there holds the following error estimate

kwd;nmax ðx; Þ wðx; Þk 6 ðs þ 1Þ1x þ

px 1 E x 1x ln x þ oð1Þ E d ðs 1Þcd ðs 1Þ

for d ! 0;

ð4:1Þ

where c is the constant in (iii) of Theorem 3.1. Proof. Deﬁne

1 wnmax ðx; yÞ :¼ pﬃﬃﬃﬃﬃﬃﬃ n ð 2p Þ

Z

iyn

e

Rn

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 sinh x jnj2 k ^ v dn; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ nmax 2 jnj2 k

ð4:2Þ

then by the Parseval formula and the triangle inequality, we have

^ d;nmax ðx; Þ wðx; ^ Þk 6 kw ^ d;nmax ðx; Þ w ^ nmax ðx; Þk þ kw ^ nmax ðx; Þ wðx; ^ Þk :¼ I1 þ I2 : kwd;nmax ðx; Þ wðx; Þk ¼ kw

ð4:3Þ

We consider (4.3) in two cases. Case 1. nmax 6 k. Note that (4.2) and the following inequality

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sin x k2 jnj2 6 x k2 jnj2 ;

for k P jnj;

ð4:4Þ

we have

2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z sinh x jnj2 k2 sin x k2 jnj2 ^ d wÞ ^ dn ¼ ^ d wÞ ^ dn 6 x2 ^ d wj ^ 2 dn 6 x2 d2 ; I21 ¼ jw qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðw qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðw 2 2 2 2 jnj6nmax jnj6nmax jnj6nmax jnj k k jnj Z

i.e.,

I1 6 xd:

ð4:5Þ

2 2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z sinhx jnj2 k2 sinhx jnj2 k2 sinhx jnj2 k2 ^ dn ¼ ^ dn þ ^ dn :¼ I2 þ I2 : I22 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ 3 4 2 2 2 jnjPnmax nmax 6jnj6k jnjPk jnj2 k jnj2 k jnj2 k Z

ð4:6Þ Due to the inequality (4.4), the triangle inequality and (3.3), we have

2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z Z sinh x jnj2 k2 sin x k2 jnj2 2 2 ^ ^ ^ I3 ¼ jwðnÞj dn qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn 6 x2 2 2 nmax 6jnj6k nmax 6jnj6k nmax 6jnj6k jnj2 k k jnj2 Z 2 2 ^ ^ ^ d ðnÞ þ ð1 v Þðw ^ w ^ d Þk2 6 x2 ððs þ 1ÞdÞ2 : jwðnÞj dn ¼ x2 kð1 v ÞwðnÞk ¼ x2 kð1 v Þw 6 x2 Z

jnjPnmax

nmax

nmax

nmax

ð4:7Þ

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Using (2.6), (2.7), the Hölder inequality, (3.3) and (3.6), we have

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 2 ^ I4 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn 2 2 jnjPk jnj k Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x jnj2 k2 ^ wðnÞ dn 6 e Z

¼

Z

jnjPk

jnjPk

¼

Z

jnjPk

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ px px jnj2 k2 ^ 2x 2ð1xÞ ^ wðnÞ ð1 þ jnj2 Þ ð1 þ jnj2 Þ jwðnÞj dn e pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x 1x px p jnj2 k2 ^ 2 2 ^ ð1 þ jnj2 Þ 1x jwðnÞj dn wðnÞ ð1 þ jnj2 Þ e

Z 6 jnjPk

6 E2x

!x Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ dn e

Z

ð1 þ jnj2 Þ

px

1x

!1x 2 ^ jwðnÞj dn

jnjPk

!1x 2px qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1x 2 2 ^ jwðnÞj dn jnj2 k

jnjPk

6 E2x

Z

ð1 þ oð1ÞÞ ln

jnjPk

E ðs 1Þcd

2x p 1x

!1x 2 ^ jwðnÞj dn

for d ! 0

!1x 2px Z E 2 ^ jwðnÞj dn for d ! 0 ðs 1Þcd jnjPnmax 2px 2ð1xÞ E ^ 6 E2x ð1 þ oð1ÞÞ ln for d ! 0 ð1 vnmax Þw ðs 1Þcd 2px E ððs þ 1ÞdÞ2ð1xÞ for d ! 0 6 E2x ð1 þ oð1ÞÞ ln ðs 1Þcd 2px E for d ! 0: ¼ ðs þ 1Þ2ð1xÞ þ oð1Þ E2x d2ð1xÞ ln ðs 1Þcd 6 E ð1 þ oð1ÞÞ ln 2x

ð4:8Þ

Combining (4.6) with (4.7) and (4.8), we have

I22 ¼ I23 þ I24 6 x2 ððs þ 1ÞdÞ2 þ ðs þ 1Þ2ð1xÞ þ oð1Þ E2x d2ð1xÞ ln ¼ ðs þ 1Þ2ð1xÞ þ oð1Þ E2x d2ð1xÞ ln

2px E ðs 1Þcd

2px E ðs 1Þcd

for d ! 0

for d ! 0

and therefore,

I2 6 ðs þ 1Þ1x þ oð1Þ Ex d1x ln

px E ðs 1Þcd

for d ! 0:

ð4:9Þ

From (4.3), (4.5) and (4.9), we obtain the following estimate for Case 1

kwd;nmax ðx; Þ wðx; Þk 6 xd þ ðs þ 1Þ1x þ oð1Þ Ex d1x ln

px E for d ! 0 ðs 1Þcd px E for d ! 0: ¼ ðs þ 1Þ1x þ oð1Þ Ex d1x ln ðs 1Þcd

ð4:10Þ

Case 2. nmax P k. Due to inequalities (2.7), (2.8) and (3.5), we have

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2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 ^ d wÞ ^ dn I21 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðw 2 jnj6nmax jnj2 k 2 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 Z Z sinh x jnj2 k2 sinh x jnj2 k2 ^ d wÞ ^ dn þ ^ d wÞ ^ dn ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðw qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðw 2 2 jnj6k k6jnj6nmax jnj2 k jnj2 k Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x jnj2 k2 ^ ^ dn 6 x2 d2 þ ðwd wÞ e k6jnj6nmax pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃZ 2 2 ^ d wj ^ 2 dn jw 6 x2 d2 þ e2x nmax k Z

Rn

p 2x E ð1 þ oð1ÞÞ d2 for d ! 0 ðs 1Þd 2x 2px 1 E E2x d2ð1xÞ ln for d ! 0 ¼ x2 d2 þ ð1 þ oð1ÞÞ ðs 1Þd s1 ! 2x 2px 1 E ¼ þ oð1Þ E2x d2ð1xÞ ln for d ! 0 ðs 1Þd s1

6 x2 d2 þ

E ðs 1Þd

ln

and therefore,

I1 6

1 s1

x

þ oð1Þ Ex d1x ln

E ðs 1Þd

px

for d ! 0:

ð4:11Þ

Meanwhile, by (2.6), (2.7), the Hölder inequality, (3.3) and (3.6), we have

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinh x jnj2 k2 2 ^ I2 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞ dn 2 jnjPnmax jnj2 k Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 x jnj2 k2 ^ wðnÞ dn 6 e Z

¼

Z

jnjPnmax

jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x 1x px p jnj2 k2 ^ 2 2 ^ wðnÞ ð1 þ jnj2 Þ ð1 þ jnj2 Þ 1x jwðnÞj dn e

Z 6 jnjPnmax

6 E2x

!x Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ dn e

Z

ð1 þ jnj2 Þ

px 1x

!1x 2 ^ jwðnÞj dn

jnjPnmax

!1x 2px qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1x 2 2 ^ jwðnÞj dn jnj2 k

jnjPnmax

6 E2x

Z

!1x 2px 1x E 2 ^ jwðnÞj dn for d ! 0 ðs 1Þcd 2px ððs þ 1ÞdÞ2ð1xÞ for d ! 0;

ð1 þ oð1ÞÞ ln

jnjPnmax

6 E2x ð1 þ oð1ÞÞ ln

E ðs 1Þcd

i.e.,

I2 6 ðs þ 1Þ1x þ oð1Þ Ex d1x ln

px E ðs 1Þcd

for d ! 0:

ð4:12Þ

Combining (4.3) with (4.11) and (4.12), we obtain the following estimate for Case 2

px 1 E x 1x ln x þ oð1Þ E d ðs 1Þd ðs 1Þ px E þ ðs þ 1Þ1x þ oð1Þ Ex d1x ln for d ! 0 ðs 1Þcd px 1 E 1x x 1x E ¼ þ ð s þ 1Þ þ oð1Þ d ln for d ! 0: ðs 1Þcd ðs 1Þx

kwd;nmax ðx; Þ wðx; Þk 6

ð4:13Þ

From (4.10) and (4.13), we obtain the ﬁnal estimate (4.1). h Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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By an argument similar to the proof of Theorem 4.1, in which the inequality (3.5) is replaced by (3.4) in Theorem 3.1, we can easily obtain. Theorem 4.2. Assume the conditions (2.3) and (2.6) with p ¼ 0 hold, if the regularization parameter nmax is chosen as the solution of Eq. (3.3), then there holds the error estimate

kwd;nmax ðx; Þ wðx; Þk 6 ðs þ 1Þ1x þ

1 Ex d1x þ oð1Þ ðs 1Þx

for

d ! 0:

ð4:14Þ

5. Numerical implementation In this section, some numerical examples are devised to verify the validity of the a posteriori Fourier method proposed in this paper. As mentioned in [1], in certain physical situations we can expect that the data function is almost zero outside a certain small domain of Rn . For simplicity (but without loss of generality), otherwise speciﬁed, we will always ﬁx the domain as

X :¼ fðx; yÞj 0 < x 6 1; jyi j 6 10; i ¼ 1; 2; . . . ; ng in these numerical experiments. The number of grids on domain X is taken to be M 2 , where M ¼ 101. Let the vector W ¼ ðW1 ; W2 ; . . . ; WM Þ represent the discrete form of the data function wðyÞ. Note that in practice, the data wðyÞ is obtained by measurement and therefore it is inevitably contaminated by measurement error, some uniformly distributed random noises e are added to W in our test examples, i.e.,

Wd ¼ W þ e randðsizeðWÞÞ: Then the noise level d can be calculated by

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u M u1 X d¼t jWi Wdi j2 M i¼1 and it is easy to see that e and d posses the same order of magnitude. In order to investigate the algorithm, we evaluate the relative error er deﬁned by

er ¼

kwðx; Þ wd;nmax ðx; Þkl2 : kwðx; Þkl2

In order to ensure the computation speed, we use the fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) techniques. The regularization parameter nmax for the a posteriori Fourier method is selected to be the solution of Eq. (3.3), i.e.,

^ d ðnÞk ¼ sd; kð1 vnmax Þw where we take s ¼ 1:1 in our computation, as suggested by Hanke and Hansen [23]. To illustrate the advantage of the a posteriori method, the comparison of numerical effect between the a posteriori and the a priori Fourier method is also taken into account in Example 5.1, where the regularization parameter for the a priori Fourier method is selected by formula (4.16) in [3]. Now we consider the following three examples. 2

Example 5.1. Let n ¼ 1; uðyÞ ¼ 0, wðyÞ ¼ ey 2 SðRÞ, where SðRÞ denotes the Schwartz function space. In this case ^ wðnÞ 2 SðRÞ decays rapidly, formula (2.1) can be used to directly calculate wðx; yÞ with exact data

1 wðx; yÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ 2p

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 jnj2 k sinh x ^ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wðnÞdn: eiyn 2 1 jnj2 k

Z

1

For simplicity, except for Fig. 3, we only consider the case of k ¼ 1. Fig. 1 gives the numerical results computed by both the a priori and the a posteriori Fourier method at x ¼ 0:1; 0:5; 0:9 with e ¼ 102 for Example 5.1. In order to make a clear comparison, the a priori bound E for the a priori method is calculated according to formula (2.3) with p ¼ 2 in [3] and the regularization parameter nmax is computed by Theorem 4.1 in [3]. Tables 1 and 2 give the comparison of relative errors between the exact solution and the a priori regularization solution and errors between the exact solution and the a posteriori regularization solution at different points for Example 5.1 with e ¼ 102 and e ¼ 103 , respectively. Fig. 1, Tables 1 and 2 show that the a posteriori Fourier method is more stable than the a priori Fourier method, and the smaller the e, the better the numerical results. Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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Fig. 1. (a)–(c) are comparisons of the exact solution and its approximation at x ¼ 0:1; 0:5; 0:9 with a priori and a posteriori choice rules for Example 5.1, respectively.

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C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 1 The comparison of relative errors for Example 5.1 with

e ¼ 102 .

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

er (a priori) er (a posteriori)

0.0202 0.0197

0.0206 0.0197

0.0214 0.0198

0.0234 0.0201

0.0282 0.0207

0.0386 0.0221

0.0596 0.0250

0.0995 0.0305

0.1734 0.0402

Table 2 The comparison of relative errors for Example 5.1 with

e ¼ 103 .

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

er (a priori) er (a posteriori)

0.0021 0.0020

0.0021 0.0020

0.0023 0.0020

0.0028 0.0020

0.0041 0.0021

0.0073 0.0022

0.0143 0.0026

0.0299 0.0032

0.0642 0.0042

Fig. 2 illustrates the impact of a priori bound on the accuracy of the approximate solution at x ¼ 0:9 with e ¼ 102 for Example 5.1. From this ﬁgure, we see that working with a wrong a priori bound may lead to a bad regularization solution. However, the a priori bound cannot be known exactly in practice, this is also a defect of any a priori regularization methods, while for the a posteriori Fourier method, there is not such a limitation. Fig. 3 compares the relative errors between the exact solution and the a posteriori regularization solution for different k at x ¼ 0:5 with e ¼ 103 for Example 5.1. This ﬁgure shows that the a posteriori Fourier method presented in this paper is effective for different wave number k. It also indicates the rationality from the aspect of direct numerical simulation that there is no restrictions on k in the theoretical analysis in Sections 3 and 4. Example 5.2. Take uðyÞ ¼ 0; wðyÞ ¼

pﬃﬃﬃ jyj ae 2 L2 ðRn Þ. For a ¼ k2 þ n, it is easy to verify that

pﬃﬃﬃ wðx; yÞ ¼ ejyj sin ax is the exact solution of problem (1.3) with weak singularity. Table 3 gives the comparison of the errors between the exact and the a posteriori regularization solution for n ¼ 1; k ¼ 1; e ¼ 102 at different x. We see that the smaller the value of x, the better the computed approximation we can get. Fig. 4 illustrates the comparisons between the exact and the a posteriori regularization solutions with three different noise levels added into the Neumann data, which shows that the smaller the noise level, the better the approximate effect. Fig. 5 gives the graphs for Example 5.2 with n ¼ 2; k ¼ 1 at x ¼ 0:2, which shows that the a posteriori Fourier method is also effective for two dimensional case. Example 5.3. It is easy to see that the function

pﬃﬃﬃ wðx; yÞ ¼ sin 2ky sinh kx

Fig. 2. The impact of a priori bound E on the a priori Fourier approximate solution at x ¼ 0:9 for Example 5.1.

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Fig. 3. The impact of wave number k on the a posteriori Fourier approximate solution at x ¼ 0:5 with

Table 3 The error between the exact and the a posteriori approximate solutions for Example 5.2 with n ¼ 1; k ¼ 1; x er

0.1 0.0665

0.2 0.0674

0.3 0.0701

0.4 0.0762

0.5 0.0874

e ¼ 103 for Example 5.1.

e ¼ 102 at different x.

0.6 0.1052

Fig. 4. The comparison of the exact and the a posteriori regularization solutions with different

0.7 0.1304

0.8 0.1639

0.9 0.2069

e at x ¼ 0:5 for Example 5.2.

pﬃﬃﬃ is the exact solution of problem (1.3) with n ¼ 1; uðyÞ 0; wðyÞ ¼ k sin 2ky. Fig. 6 gives the comparison of the exact solution and the a posteriori approximate solution for Example 5.3 with k ¼ 1 at x ¼ 0:6 for different noise level. Although wðx; Þ and wðÞ do not belong to L2 ðRÞ for this example, i.e., the a priori bound (2.6) cannot be satisﬁed for any p P 0, Fig. 6 shows that the a posteriori Fourier method still has good computational effect.

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Fig. 5. Example 5.2 with n ¼ 2 at x ¼ 0:2: (a) the exact solution; (b) the a posteriori Fourier regularization solution with e ¼ 102 ; (c) the error graph with e ¼ 102 .

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Fig. 6. The comparison of the exact and the a posteriori regularization solutions with different

e at x ¼ 0:6 for Example 5.3.

6. Conclusion In this paper, the Cauchy problem for the Helmholtz equation with only inhomogeneous Neumann data (1.3) is specially considered. In order to facilitate the theoretical analysis, a new and nonstandard a priori bound is introduced, which is a reﬁnement of the standard Hp ðRn Þ-norm and can better characterize the essential nature of ill-posedness of this problem. By utilizing the a priori bound, a conditional stability result is proved, which also ensures the rigor of the a posteriori choice rule of the regularization parameter. The most important task of this paper is that a very simple and rather effective a posteriori Fourier regularization method for solving problem (1.3) is established. We ﬁnd some crucial information about the regularization parameter hidden in the a posteriori choice rule, which help us to get some more accurate logarithmic-Hölder type error estimates. These results further improve and perfect the theory of the a posteriori Fourier method. In addition, for all the theoretical analysis in this paper, we do not impose any restrictions on the wave number k in the Helmholtz equation. Meanwhile, numerical examples show that though both the a priori and the a posteriori Fourier methods are effective regularization methods for solving the problem considered in this paper, the computational result of the a priori Fourier method is strongly affected by the a priori bound. Working with a wrong a priori bound may lead to bad regularization solution. The a posteriori Fourier method presented in this paper is a more practical method. Except for the theoretical analysis, both the choice of the regularization parameter and the numerical computation completely do not need the value of the a priori bound. In addition, the numerical effect of the a posteriori Fourier method is better than the a priori one. Appendix A. Proof of Theorem 3.1 For the proof we use a known lemma. Lemma A.1 [24]. Let the function f ðkÞ : ½0; a ! R be given by

c 1 f ðkÞ ¼ kb d ln k with a constant c 2 R and positive constants a < 1; b and d, then for the inverse function f

f

1

1

ðkÞ ¼ kb

c d 1 b ln ð1 þ oð1ÞÞ for k ! 0: b k

ðA:1Þ 1

ðkÞ, we have

ðA:2Þ

Proof of Theorem 3.1. Proof of part (i). Due to the condition (2.6) for p ¼ 0, we have

Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

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C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

Z

^ ¼ kð1 vnmax ÞwðnÞk

¼

jnjPnmax

6

!12 2 ^ jwðnÞj dn

Z jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z

max e

jnj2 k2

jnjPnmax

jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jnj2 k2 ^ 2 2 jnj2 k2 wðnÞ e dn e

!12

!12 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 jnj2 k2 ^ 2 wðnÞ dn 6 e nmax k E: e

ðA:3Þ

Meanwhile, in view of Eq. (3.3) and the triangle inequality, we have

^ ^ ^ d ðnÞ þ w ^ d ðnÞk P kð1 v Þw ^ d ðnÞk kð1 v ÞðwðnÞ ^ ^ d ðnÞÞk ¼ kð1 vnmax ÞðwðnÞ w w kð1 vnmax ÞwðnÞk nmax nmax P ðs 1Þd:

ðA:4Þ

Combining (A.3) with (A.4), we have

e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 2 nmax k

E P ðs 1Þd:

This just is the estimate (3.4). Proof of part (ii). Similar to the proof in (i), we have

^ ¼ kð1 vnmax ÞwðnÞk

Z

!12 2 ^ jwðnÞj dn

¼

jnjPnmax

Z jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p p 2 2 jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ ð1 þ jnj2 Þ e2 jnj k dn e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp Z 2 2 1 þ jnj2 6 max e jnj k jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 2 2 6 e nmax k E: n2max k

jnjPnmax

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p jnj2 k2 ^ 2 wðnÞ ð1 þ jnj2 Þ dn e

!12

!12

ðA:5Þ

Meanwhile, the estimate (A.4) holds, combining (A.5) with (A.4), we have

e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 n2max k2 E P ðs 1Þd; n2max k

i.e.,

e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp ðs 1Þd 2 n2max k2 P : n2max k E

ðA:6Þ

Let

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 jnj2 k ¼ s

ðA:7Þ

and deﬁne the function

hðsÞ ¼ es sp ;

s P 0; p > 0:

ðA:8Þ

Instead of inequality (A.6), we consider the following inequality

hðsÞ ¼ es sp P

ðs 1Þd : E

ðA:9Þ

Note that 0

h ðsÞ ¼ ðp þ sÞes sp1 < 0; function hðsÞ is strictly decreasing. If s is the solution of the following equation

es sp ¼

ðs 1Þd E

ðA:10Þ

or equivalently, n is the solution of equation

e

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðs 1Þd 2 n2 k2 ¼ ; n2 k E

ðA:11Þ

then the solution s of inequality (A.9) will satisfy

s 6 s ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 n2 k ;

especially, for inequality (A.6), there holds

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C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 n2max k 6 s :

ðA:12Þ

Now let us consider solving Eq. (A.10). In order to apply Lemma A.1 conveniently, let

es ¼ k; thus s ¼

ln 1k,

ðA:13Þ

and k 2 ð0; 1Þ, Eq. (A.10) becomes

p 1 ðs 1Þd k ln ¼ : k E

ðA:14Þ

The left hand side of Eq. (A.14) just is the form of f ðkÞ given by (A.1) with b ¼ 1; d ¼ 1; c ¼ p. From Lemma A.1 we obtain an asymptotic expression of the solution of Eq. (A.14),

k ¼

p ðs 1Þd E ln ð1 þ oð1ÞÞ for d ! 0: E ðs 1Þd

Due to (A.13), we know

k ¼ es ¼ e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 n k

¼

p ðs 1Þd E ln ð1 þ oð1ÞÞ for d ! 0: E ðs 1Þd

p h p i E E , here A ¼ AðdÞ is an inﬁnitely small quantity when d ! 0 and Denote A :¼ ðs1Þd ln ðs1Þd and e :¼ oð1Þ ðs1Þd ln ðs1Þd E E e ¼ eðdÞ is a high order inﬁnitely small quantity with respect to A. Thus the above formula can be rewritten as

e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 n k

¼Aþe

and

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 e n k ¼

1 : Aþe

ðA:15Þ

Note that 1 Aþe 1 A

¼

A 1 ¼ ! 1 for d ! 0; A þ e 1 þ Ae

therefore,

1 1 ¼ ð1 þ oð1ÞÞ Aþe A

for d ! 0;

i.e.,

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 1 e n k ¼ ð1 þ oð1ÞÞ A

for d ! 0:

and

p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 1 E E for d ! 0: e nmax k 6 e n k ¼ ð1 þ oð1ÞÞ ¼ ð1 þ oð1ÞÞ ln A ðs 1Þd ðs 1Þd Conclusion (ii) is proved. Proof of part (iii). From (A.15), we know

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 n2 k ¼ ln

1 Aþe

and

1 1 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ¼ : lnðA þ eÞ 2 ln Aþ1 e 2 n k Note that

ln A þ lnð1 þ Ae Þ ln 1 þ Ae lnðA þ eÞ ln A 1 þ Ae ¼ ¼ ¼1þ ! 1 for d ! 0 ln A ln A ln A ln A and therefore there holds

lnðA þ eÞ ¼ ð1 þ oð1ÞÞ ln A for d ! 0: So Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

1 1 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ¼ ð1 þ oð1ÞÞ ln A ð1 þ oð1ÞÞ ln A1 2 2 n k

17

for d ! 0;

i.e.,

p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E E 2 for d ! 0: ln n2 k ¼ ð1 þ oð1ÞÞ ln ðs 1Þd ðs 1Þd Deﬁne the function

HðnÞ ¼ hðsðnÞÞ ¼ e

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 jnj2 k2 jnj2 k ;

then

Hðn Þ ¼ e

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 n2 k2 n2 k

and

Hðnmax Þ ¼ e

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 n2max k2 ; n2max k

where n and nmax are the solution of Eqs. (A.11) and (3.3), respectively. For jnj P n , there holds

p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E E 2 2 for d ! 0 ln jnj2 k P n2 k ¼ ð1 þ oð1ÞÞ ln ðs 1Þd ðs 1Þd and therefore

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 jnj2 k 6 ð1 þ oð1ÞÞ ln

p p E E ln ðs 1Þd ðs 1Þd

for d ! 0:

ðA:16Þ

Note that

p p 0 p 1p E E E E ð1 þ oð1ÞÞ ln ðs1Þd ln ðs1Þd Bð1 þ oð1ÞÞ ln ðs1Þd ln ðs1Þd C ¼@ p A ! 1 for d ! 0 p p E E E E ln ðs1Þd ln ðs1Þd ln ðs1Þd ln ðs1Þd and

0

1p

0 1p p p @ 1 p A E E E þln ln E E lnðs1Þd ln ðs1Þd ln ðs1Þd ln ðs1Þd ðs1Þd B C ¼ ¼@ p p p A ! 1 for d ! 0: E E E E ln ðs1Þd ln ðs1Þd ln ðs1Þd þ ln ln ðs1Þd So,

p p p p p p E E E E E E ð1 þ oð1ÞÞln ðs1Þd ln ðs1Þd ð1 þ oð1ÞÞln ðs1Þd ln ðs1Þd ln ðs1Þd ln ðs1Þd ¼ ! 1 for d ! 0: p p p p E E E E ln ðs1Þd ln ðs1Þd ln ðs1Þd ln ðs1Þd Thus, formula (A.16) can be written as

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 6 ð1 þ oð1ÞÞ ln jnj2 k

E ðs 1Þd

p for d ! 0:

ðA:17Þ

For nmax 6 jnj 6 n , note that there is a constant c > 1, such that

Hðnmax Þ 6 cHðn Þ ¼

ðs 1Þcd E

and therefore

HðnÞ 6

ðs 1Þcd ; E

8 jnj 2 ½nmax ; n :

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C.-L. Fu et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

Denote n# is the solution of the following equation

HðnÞ ¼

ðs 1Þcd ; E

then similar to the above deduction, we know for nmax 6 jnj 6 n , there holds

p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E E 2 jnj2 k P ð1 þ oð1ÞÞ ln for d ! 0 ln ðs 1Þcd ðs 1Þcd and

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃp 2 6 ð1 þ oð1ÞÞ ln jnj2 k

E ðs 1Þcd

p for d ! 0:

ðA:18Þ

Combining (A.17) with (A.18), and noting c > 1, we obtain the ﬁnal estimate (3.6) for jnj P nmax . References [1] T. Regin´ska, K. Regin´ski, Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Prob. 22 (2006) 975–989. [2] V. Isakov, Inverse Problems for Partial Differential Equation, Springer, New York, 1998. [3] C.L. Fu, X.L. Feng, Z. Qian, The Fourier regularization for solving the Cauchy problem for the Helmholtz equation, Appl. Numer. Math. 59 (2009) 2625– 2640. [4] X.T. Xiong, C.L. Fu, Two approximate methods of a Cauchy problem for the Helmholtz equation, Comput. Appl. Math. 26 (2007) 285–307. [5] T. Regin´ska, U. Tautenhahn, Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation, Numer. Funct. Anal. Optim. 30 (9–10) (2009) 1065–1097. [6] H.H. Qin, T. Wei, Modiﬁed regularization method for the Cauchy problem of the Helmholtz equation, Appl. Math. Model. 33 (2009) 2334–2348. [7] H.H. Qin, T. Wei, R. Shi, Modiﬁed Tikhonov regularization method for the Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math. 224 (2009) 39–53. [8] A.L. Qian, X.T. Xiong, Y.J. Wu, On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math. 233 (2010) 1969–1979. [9] X.T. Xiong, A regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math. 233 (2010) 1723–1732. [10] C.L. Fu, Z. Qian, Numerical pseudodifferential operator and Fourier regularization, Adv. Comput. Math. 33 (4) (2010) 449–470. [11] C.L. Fu, Y.X. Zhang, H. Cheng, Y.J. Ma, The a posteriori Fourier method for solving ill-posed problems, Inverse Prob. 28 (2012) 095002 (26pp). [12] X.L. Feng, C.L. Fu, H. Cheng, A regularization method for solving the Cauchy problem for the Helmholtz equation, Appl. Math. Model. 35 (2011) 3301– 3315. [13] L. Eldén, F. Berntsson, T. Regin´ska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (6) (2000) 2187–2205. [14] C.L. Fu, Y.J. Ma, H. Cheng, Y.X. Zhang, The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data, Appl. Math. Model. 37 (2013) 7764–7777. [15] H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Boston, 1996. [16] D.N. Hào, H. Sahli, On a class of severely ill-posed problems, Vietnam J. Math. 32 (2004) 143–152. [17] C.L. Fu, X.T. Xiong, Z. Qian, Fourier regularization for a backward heat equation, J. Math. Anal. Appl. 331 (1) (2007) 472–480. [18] C.L. Fu, H.F. Li, Z. Qian, X.T. Xiong, Fourier regularization method for solving a Cauchy problem for the Laplace equation, Inverse Prob. Sci. Eng. 16 (2) (2008) 159–169. [19] C.L. Fu, F.F. Dou, X.L. Feng, Z. Qian, A simple regularization method for stable analytic continuation, Inverse Prob. 24 (2008) 065003 (15pp). [20] F.F. Dou, C.L. Fu, F.L. Yang, Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation, J. Comput. Appl. Math. 230 (2) (2009) 728–737. [21] V.A. Morozov, Choice of parameter for the solution of functional equations by the regularization method, Sov. Math. Dokl. 8 (1967) 1000–1003. [22] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996. [23] M. Hanke, D.C. Hansen, Regularization methods for large-scale problems, Surv. Math. Ind. 3 (1993) 253–315. [24] U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim. 19 (1998) 377–398.

Please cite this article in press as: C.-L. Fu et al., A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2014.12.030

A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data

A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data

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