International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 435468 | 18 pages | https://doi.org/10.5402/2012/435468

A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

Academic Editor: L. Marin
Received16 Aug 2011
Accepted28 Sep 2011
Published04 Jan 2012

Abstract

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 Earthā€™s subsurface [3]. 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 [4]. 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 [5]. 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 ā€œstripā€ 0<š‘„<1 as follows:Ī”š‘¢(š‘„,š‘¦)+š‘˜2š‘¢(š‘„,š‘¦)=0,š‘„āˆˆ(0,1),š‘¦āˆˆā„š‘›,š‘›ā‰„1,š‘¢(0,š‘¦)=š‘”(š‘¦),š‘¦āˆˆā„š‘›,š‘¢š‘„(0,š‘¦)=0,š‘¦āˆˆā„š‘›,(1.1) where Ī”=šœ•2/šœ•š‘„2+āˆ‘š‘›š‘–=1šœ•2/šœ•š‘¦2š‘– is an š‘›+1 dimensional Laplace operator. We want to determine the solution š‘¢(š‘„,š‘¦) for 0<š‘„ā‰¤1 from the data š‘”(š‘¦). 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 [8], Johansson and Martin [9], and Marin et al. [10ā€“14].

Let š’® be the Schwartz space over ā„š‘›, and let š’®ā€² be its dual (the space of tempered distributions). Let īš‘“ denote the Fourier transform of function š‘“(š‘¦)āˆˆš’® defined byī1š‘“(šœ‰)=(2šœ‹)š‘›/2ī€œā„š‘›š‘’āˆ’š‘–šœ‰ā‹…š‘¦ī€·šœ‰š‘“(š‘¦)š‘‘š‘¦,šœ‰=1,ā€¦,šœ‰š‘›ī€øī€·š‘¦,š‘¦=1,ā€¦,š‘¦š‘›ī€ø,(1.2) while the Fourier transform of a tempered distribution š‘“āˆˆš’®ā€² is defined byī‚€īī‚=ī‚€īšœ™ī‚š‘“,šœ™š‘“,,āˆ€šœ™āˆˆš’®.(1.3) In this paper, we will consider functions depending on the variables š‘„āˆˆ[0,1], š‘¦āˆˆā„š‘›.

For š‘ āˆˆā„, the Sobolev space š»š‘ (ā„š‘›) consists of all tempered distributions š‘“(š‘¦)āˆˆš’®ā€², for which īš‘“(šœ‰)(1+|šœ‰|2)š‘ /2 is a function in šæ2(ā„š‘›). The norm on this space is given byā€–š‘“ā€–š»š‘ ī‚µī€œāˆ¶=ā„š‘›||ī||š‘“(šœ‰)2||šœ‰||(1+2)š‘ ī‚¶š‘‘šœ‰1/2.(1.4)

We assume there exists a unique solution š‘¢(š‘„,š‘¦) of problem (1.1), which satisfies the problem in the classical sense and š‘”(ā‹…), š‘¢(š‘„,ā‹…)āˆˆšæ2(ā„š‘›). Applying the Fourier transform technique to problem (1.1) with respect to the variable š‘¦ yields the following problem in the frequency space:Ģ‚š‘¢š‘„š‘„ī‚€š‘˜(š‘„,šœ‰)+2āˆ’||šœ‰||2ī‚Ģ‚š‘¢(š‘„,šœ‰)=0,š‘„āˆˆ(0,1),šœ‰āˆˆā„š‘›,š‘›ā‰„1,Ģ‚š‘¢(0,šœ‰)=Ģ‚š‘”(šœ‰),šœ‰āˆˆā„š‘›,Ģ‚š‘¢š‘„(0,šœ‰)=0,šœ‰āˆˆā„š‘›.(1.5) It is easy to obtain the solution of problem (1.5) (if exists) has the formī‚µš‘„ī”Ģ‚š‘¢(š‘„,šœ‰)=cosh||šœ‰||2āˆ’š‘˜2ī‚¶Ģ‚š‘”(šœ‰),(1.6) or equivalently, the solution of problem (1.1) has the representation1š‘¢(š‘„,š‘¦)=(2šœ‹)š‘›/2ī€œā„š‘›š‘’š‘–šœ‰ā‹…š‘¦ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶Ģ‚š‘”(šœ‰)š‘‘šœ‰.(1.7) Since āˆšcosh(š‘„|šœ‰|2āˆ’š‘˜2) increases rapidly with exponential order as |šœ‰|ā†’āˆž, the Fourier transform of the exact data š‘”(š‘¦) must decay rapidly. However, in practice, the data at š‘„=0 is often obtained on the basis of reading of physical instrument which is denoted by š‘”š‘š. We assume that š‘”(ā‹…) and š‘”š‘š(ā‹…) satisfyā€–ā€–š‘”(ā‹…)āˆ’š‘”š‘šā€–ā€–(ā‹…)š»š‘Ÿā‰¤š›æ.(1.8) Since š‘”š‘š(ā‹…) belong to šæ2(ā„š‘›)āŠ‚š»š‘Ÿ(ā„š‘›) for š‘Ÿā‰¤0, š‘Ÿ should not be positive. A small perturbation in the data š‘”(š‘¦) may cause a dramatically large error in the solution š‘¢(š‘„,š‘¦) for 0<š‘„ā‰¤1. 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 knowī‚µī”Ģ‚š‘¢(1,šœ‰)=cosh||šœ‰||2āˆ’š‘˜2ī‚¶Ģ‚š‘”(šœ‰).(1.9) Since the convergence rates can only be given under a priori assumptions on the exact solution [15], we will formulate such an a priori assumption in terms of the exact solution at š‘„=1 by consideringā€–š‘¢(1,ā‹…)ā€–š»š‘ ā‰¤šø.(1.10)

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 [18] and multidimensional [19] cases, and so forth. In this paper we propose a similar wavelet method as suggested in [19] 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 Ī¦ be Meyerā€™s orthonormal scaling function in š‘› dimensions. This function is constructed from the one-dimensional scaling functions in the following way. Let šœ™(š‘„) and šœ“(š‘„) be the Meyer scaling and wavelet function in one dimension defined by their Fourier transform in [20] which satisfyīī‚ƒāˆ’4suppšœ™=34šœ‹,3šœ‹ī‚„,ī‚ƒāˆ’8suppīšœ“=32šœ‹,āˆ’3šœ‹ī‚„āˆŖī‚ƒ23,83šœ‹ī‚„.(2.1) It can be proved (cf. [20]) that the set of functionsšœ“š‘—š‘˜(š‘„)=2š‘—/2šœ“ī€·2š‘—ī€øš‘„āˆ’š‘˜,š‘—,š‘˜āˆˆā„¤,(2.2) is an orthonormal basis of šæ2(ā„). Consequently, the MRA {š‘‰š‘—}š‘—āˆˆā„¤ of Meyer is generated byš‘‰š‘—={šœ™š‘—š‘˜,š‘˜āˆˆā„¤},šœ™š‘—š‘˜āˆ¶=2š‘—/2šœ™ī€·2š‘—ī€øī‚€īšœ™š‘„āˆ’š‘˜,š‘—,š‘˜āˆˆā„¤,suppš‘—š‘˜ī‚=ī‚†||šœ‰||ā‰¤4šœ‰;3šœ‹2š‘—ī‚‡.(2.3)

For the construction of an š‘›-dimensional MRA, we take tensor products of the spaces š‘‰š‘— (see [21, 22]). Then the scaling function Ī¦ is given byĪ¦(š‘„)=š‘›ī‘š‘˜=1šœ™ī€·š‘„š‘˜ī€ø,š‘„āˆˆā„š‘›,(2.4) and any basis function ĪØ in š‘Šš½ can be written in the formĪØ(š‘„)=2š‘›š½/2šœ“ī€·2š½š‘„š‘–āˆ’š‘˜š‘–ī€øā‹…ī‘š‘šā‰ š‘–šœƒš‘šī€·2š½š‘„š‘šāˆ’š‘˜š‘šī€ø,š‘„āˆˆā„š‘›,(2.5) where š‘˜āˆˆā„¤š‘›, and for any š‘šāˆˆ{1,ā€¦,š‘›}, šœƒš‘š stands for šœ™ or šœ“. Hence we obtain from (2.1) thatī‚€īĪ¦ī‚=ī‚ƒāˆ’4supp34šœ‹,3šœ‹ī‚„š‘›,ī(2.6)š‘“(šœ‰)=0forā€–šœ‰ā€–āˆžā‰¤23šœ‹2š½,š‘“āˆˆš‘Šš½,š½āˆˆā„•.(2.7) The orthogonal projection on the space š‘‰š½ is defined byš‘ƒš½ī“š‘“āˆ¶=š‘˜āˆˆā„¤š‘›ī€·š‘“,Ī¦š½,š‘˜ī€øĪ¦š½,š‘˜,(2.8) while š‘„š½š‘“ denotes the orthogonal projection of a function š‘“ on the wavelet space š‘Šš½ with š‘‰š½+1=š‘‰š½āŠ•š‘Šš½. (In many contexts one will find more than one detailed space š‘Šš½, that is, š‘‰š½+1=š‘‰š½āŠ•š‘Š1,š½āŠ•š‘Š2,š½āŠ•ā‹Æ. Here, the space š‘Šš½ is simply defined as the orthogonal complement of š‘‰š½ in š‘‰š½+1).

LetĪ©š½āˆ¶=2š½ī‚ƒāˆ’232šœ‹,3šœ‹ī‚„š‘›.(2.9) Setting Ī“š½āˆ¶=ā„š‘›ā§µĪ©š½, together with (2.6), it follows for š½āˆˆā„• thatī‚暝‘ƒš½š‘“(šœ‰)=0foršœ‰āˆˆĪ“š½+1,ī€·ī€·š¼āˆ’š‘ƒš½ī€øš‘“ī€øĢ‚īƒ³š‘„(šœ‰)=š½š‘“(šœ‰)foršœ‰āˆˆĪ©š½+1.(2.10) We introduce the operator š‘€š½ which is defined by the equationīƒ³š‘€š½ī€·š‘“āˆ¶=1āˆ’šœ’š½ī€øīš‘“,š½āˆˆā„•,(2.11) where šœ’š½ denotes the characteristic function of the cube Ī©š½. From (2.7) it follows that any basis function ĪØ in š‘Šš‘—, š‘—ā‰„š½, satisfiesīĪØ(šœ‰)=0,šœ‰āˆˆĪ©š½,(2.12) and we obtainī‚€īīĪØī‚=ī‚€ī€·(š‘“,ĪØ)=š‘“,1āˆ’šœ’š½ī€øīīĪØī‚=ī€·š‘€š‘“,š½ī€øš‘“,ĪØ.(2.13) And it follows for š½āˆˆā„• thatš‘„š½=š‘„š½š‘€š½,š¼āˆ’š‘ƒš½=ī€·š¼āˆ’š‘ƒš½ī€øš‘€š½.(2.14)

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 [19, 23]). Let {š‘‰š½}š½āˆˆā„¤ be an m-regular MRA, and let š‘Ÿ,š‘ āˆˆā„ be such that āˆ’š‘š<š‘Ÿ<š‘ <š‘š. Then for each function š‘“āˆˆš»š‘ (ā„š‘›) and š½āˆˆā„•, the following inequality holds: ā€–ā€–š‘“āˆ’š‘ƒš½š‘“ā€–ā€–š»š‘Ÿā‰¤š¶12āˆ’š½(š‘ āˆ’š‘Ÿ)ā€–š‘“ā€–š»š‘Ÿ.(3.1)

Lemma 3.2 (see [19]). Let {š‘‰š½}š½āˆˆā„¤ be Meyerā€™s (tensor-) MRA, and suppose š½āˆˆā„•, š‘Ÿāˆˆā„. Then for all š‘“āˆˆš‘‰š½, one has ā€–ā€–ā€–šœ•š‘™šœ•š‘„š‘™š‘–š‘“ā€–ā€–ā€–š»š‘Ÿā‰¤š¶22(š½āˆ’1)š‘™ā€–š‘“ā€–š»š‘Ÿ,š‘–=1,ā€¦,š‘›,š‘™āˆˆā„•.(3.2)

Define an operator š‘‡š‘„āˆ¶š‘”(š‘¦)ā†¦š‘¢(š‘„,š‘¦) by (1.6), that is,š‘‡š‘„š‘”=š‘¢(š‘„,š‘¦),0<š‘„ā‰¤1,(3.3) or equivalently,ī‚暝‘‡š‘„ī‚µš‘„ī”š‘”(šœ‰)=cosh||šœ‰||2āˆ’š‘˜2ī‚¶Ģ‚š‘”(šœ‰),0<š‘„ā‰¤1.(3.4) Then we have

Theorem 3.3. Let {š‘‰š½}š½āˆˆā„¤ be Meyerā€™s MRA and suppose š‘Ÿāˆˆā„ and š½āˆˆā„• which satisfies 2š½>š‘˜, 0ā‰¤š‘„ā‰¤1. Then for all š‘“āˆˆš‘‰š½, one has ā€–ā€–š‘‡š‘„š‘“ā€–ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’š‘„āˆš22(š½āˆ’1)āˆ’š‘˜2ī‚+1ā€–š‘“ā€–š»š‘Ÿ.(3.5)

Proof. For š‘“āˆˆš‘‰š½, by definition (1.4) and formula (3.4), from Lemma 3.2, we have ā€–ā€–š‘‡š‘„š‘“ā€–ā€–š»š‘Ÿ=īƒ©ī€œā„š‘›||||ī”cosh(š‘„||šœ‰||2āˆ’š‘˜2)īš‘“||||2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2ā‰¤āŽ›āŽœāŽœāŽœāŽœāŽœāŽī€œ|šœ‰|>š‘˜||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī€·š‘„||šœ‰||ī€øī€·š‘„||šœ‰||ī€øīš‘“||||||||coshcosh2ī‚€||šœ‰||1+2ī‚š‘ŸāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ š‘‘šœ‰1/2+īƒ©ī€œ|šœ‰|ā‰¤š‘˜||||ī‚µš‘„ī”cosš‘˜2āˆ’||šœ‰||2ī‚¶īš‘“||||2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2ā‰¤supšœ‰āˆˆĪ©š½+1||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī€·š‘„||šœ‰||ī€ø||||||||ī‚µī€œcosh|šœ‰|>š‘˜||ī€·š‘„||šœ‰||ī€øīš‘“||cosh2ī‚€||šœ‰||1+2ī‚š‘Ÿī‚¶š‘‘šœ‰1/2+ā€–š‘“ā€–š»š‘Ÿā‰¤2supšœ‰āˆˆĪ©š½+1š‘’āˆšš‘„(|šœ‰|2āˆ’š‘˜2āˆ’|šœ‰|)āŽ›āŽœāŽœāŽī€œ|šœ‰|>š‘˜|||||āˆžī“š‘™=0š‘„2š‘™(2š‘™)!|šœ‰|2š‘™īš‘“|||||2ī‚€||šœ‰||1+2ī‚š‘ŸāŽžāŽŸāŽŸāŽ š‘‘šœ‰1/2+ā€–š‘“ā€–š»š‘Ÿā‰¤2š¶3š‘’āˆšš‘„(22(š½āˆ’1)āˆ’š‘˜2āˆ’2š½āˆ’1)āˆžī“š‘™=0š‘„2š‘™ā€–ā€–(2š‘™)!(Ī”š‘¦)š‘™š‘“ā€–ā€–š»š‘Ÿ+ā€–š‘“ā€–š»š‘Ÿā‰¤īƒ©š¶4š‘’āˆšš‘„(22(š½āˆ’1)āˆ’š‘˜2āˆ’2š½āˆ’1)āˆžī“š‘™=0š‘„2š‘™(2š‘™)!ā‹…š‘›22(š½āˆ’1)š‘™īƒŖ+1ā€–š‘“ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’āˆšš‘„(22(š½āˆ’1)āˆ’š‘˜2āˆ’2š½āˆ’1)ī€·coshš‘„2š½āˆ’1ī€øī‚+1ā€–š‘“ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’š‘„āˆš22(š½āˆ’1)āˆ’š‘˜2ī‚+1ā€–š‘“ā€–š»š‘Ÿ.(3.6)

Since the Cauchy data are given inexactly by š‘”š‘š, we need a stable algorithm to approximate the solution of (1.1). Our method is as follows. Consider the operatorš‘‡š‘„,š½āˆ¶=š‘ƒš½š‘‡š‘„š‘ƒš½,(3.7) and show that it approximates š‘‡š‘„ in a stable way for an appropriate choice for š½āˆˆā„• depending on š›æ and šø. By the triangle inequality we knowā€–ā€–š‘‡š‘„š‘”āˆ’š‘‡š‘„,š½š‘”š‘šā€–ā€–š»š‘Ÿā‰¤ā€–ā€–(š‘‡š‘„āˆ’š‘‡š‘„,š½ā€–ā€–)š‘”š»š‘Ÿ+ā€–ā€–š‘‡š‘„,š½ī€·š‘”āˆ’š‘”š‘šī€øā€–ā€–š»š‘Ÿ.(3.8)

From (1.8) and Theorem 3.3, the second term on the right-hand side of (3.8) satisfiesā€–ā€–š‘‡š‘„,š½ī€·š‘”āˆ’š‘”š‘šī€øā€–ā€–š»š‘Ÿ=ā€–ā€–š‘ƒš½š‘‡š‘„š‘ƒš½ī€·š‘”āˆ’š‘”š‘šī€øā€–ā€–š»š‘Ÿā‰¤ā€–ā€–š‘‡š‘„š‘ƒš½(š‘”āˆ’š‘”š‘š)ā€–ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’š‘„āˆš22(š½āˆ’1)āˆ’š‘˜2ī‚+1š›æ.(3.9)

For the first one we haveā€–ā€–ī€·š‘‡š‘„āˆ’š‘‡š‘„,š½ī€øš‘”ā€–ā€–š»š‘Ÿā‰¤ā€–ā€–ī€·š¼āˆ’š‘ƒš½ī€øš‘‡š‘„š‘”ā€–ā€–š»š‘Ÿ+ā€–ā€–š‘ƒš½š‘‡š‘„ī€·š¼āˆ’š‘ƒš½ī€øš‘”ā€–ā€–š»š‘Ÿ.(3.10)

By Lemma 3.1, (1.9), (1.10), (2.14), and (3.4), we getā€–ā€–ī€·š¼āˆ’š‘ƒš½ī€øš‘‡š‘„š‘”ā€–ā€–š»š‘Ÿ=ā€–ā€–ī€·š¼āˆ’š‘ƒš½ī€øš‘€š½š‘‡š‘„š‘”ā€–ā€–š»š‘Ÿā‰¤š¶12āˆ’š½(š‘ āˆ’š‘Ÿ)ā€–ā€–š‘€š½š‘‡š‘„š‘”ā€–ā€–š»š‘ =š¶12āˆ’š½(š‘ āˆ’š‘Ÿ)āŽ›āŽœāŽœāŽœāŽœāŽœāŽī€œĪ“š½||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī‚µī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||||||š‘¢(1,ā‹…)2ī‚€||šœ‰||1+2ī‚š‘ āŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ 1/2ā‰¤š¶12āˆ’š½(š‘ āˆ’š‘Ÿ)supšœ‰āˆˆĪ“š½||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī‚µī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||||||ā‹…ā€–š‘¢(1,ā‹…)ā€–š»š‘ ā‰¤2š¶1ī‚µī‚€32šœ‹2š½ī‚2āˆ’š‘˜2ī‚¶āˆ’(š‘ āˆ’š‘Ÿ)/2š‘’ī”āˆ’(1āˆ’š‘„)((2/3)šœ‹2š½)2āˆ’š‘˜2šøā‰¤2š¶1ī€·22š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆšāˆ’(1āˆ’š‘„)22š½āˆ’š‘˜2šø.(3.11) On the other hand, due to (2.10), we knowā€–ā€–š‘ƒš½š‘‡š‘„ī€·š¼āˆ’š‘ƒš½ī€øš‘”ā€–ā€–š»š‘Ÿā‰¤ā€–ā€–š‘‡š‘„ī€·š¼āˆ’š‘ƒš½ī€øš‘”ā€–ā€–š»š‘Ÿā‰¤īƒ©ī€œĪ©š½+1||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī€·š‘„š½š‘”ī€øĢ‚||||(šœ‰)2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2+īƒ©ī€œĪ“š½+1||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||Ģ‚š‘”(šœ‰)2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2=āˆ¶š¼1+š¼2.(3.12) We estimate the two parts at the right-hand side of (3.12) separately. For š¼2 we haveš¼2=īƒ©ī€œĪ“š½+1||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||Ģ‚š‘”(šœ‰)2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2=āŽ›āŽœāŽœāŽœāŽœāŽœāŽī€œĪ“š½+1||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī‚µī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||||||š‘¢(1,ā‹…)2ī‚€||šœ‰||1+2ī‚š‘ŸāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ š‘‘šœ‰1/2ā‰¤supšœ‰āˆˆĪ“š½+1||||||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī‚µī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶1ī‚€||šœ‰||1+2ī‚(š‘ āˆ’š‘Ÿ)/2||||||||Ɨīƒ©ī€œĪ“š½+1||||ī‚€||šœ‰||š‘¢(1,ā‹…)1+2ī‚š‘ /2||||2īƒŖš‘‘šœ‰1/2ī‚€4ā‰¤2ā‹…3šœ‹2š½ī‚āˆ’(š‘ āˆ’š‘Ÿ)š‘’ī”āˆ’(1āˆ’š‘„)š‘›((4/3)šœ‹2š½)2āˆ’š‘˜2ā‹…ā€–š‘¢(1,ā‹…)ā€–š»š‘ ā‰¤2ā‹…(22š½āˆ’š‘˜2)āˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆšāˆ’(1āˆ’š‘„)22š½āˆ’š‘˜2šø.(3.13) Now we turn to š¼1. There holdsš¼1=īƒ©ī€œĪ©š½+1||||ī‚µš‘„ī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶ī€·š‘„š½š‘”ī€øĢ‚||||(šœ‰)2ī‚€||šœ‰||1+2ī‚š‘ŸīƒŖš‘‘šœ‰1/2ā‰¤ā€–ā€–š‘‡š‘„š‘„š½š‘”ā€–ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’š‘„āˆš22š½āˆ’š‘˜2ī‚ā€–ā€–š‘„+1š½š‘”ā€–ā€–š»š‘Ÿ,(3.14) since š‘„š½š‘”āˆˆš‘‰š½+1. Furthermore, from (2.14), it follows thatā€–ā€–š‘„š½š‘”ā€–ā€–š»š‘Ÿ=ā€–ā€–š‘„š½š‘€š½š‘”ā€–ā€–š»š‘Ÿā‰¤ā€–ā€–š‘€š½š‘”ā€–ā€–š»š‘Ÿ=ī‚µī€œĪ“š½||||Ģ‚š‘”(šœ‰)2ī‚€||šœ‰||1+2ī‚š‘Ÿī‚¶š‘‘šœ‰1/2=āŽ›āŽœāŽœāŽœāŽœāŽœāŽī€œĪ“š½||||||||Ģ‚š‘¢(1,šœ‰)ī‚µī”cosh||šœ‰||2āˆ’š‘˜2ī‚¶||||||||2ī‚€||šœ‰||1+2ī‚š‘ŸāŽžāŽŸāŽŸāŽŸāŽŸāŽŸāŽ š‘‘šœ‰1/2ā‰¤2ā‹…2āˆ’š½(š‘ āˆ’š‘Ÿ)š‘’āˆ’ī”š‘›((2/3)šœ‹2š½)2āˆ’š‘˜2ā€–š‘¢(1,ā‹…)ā€–š»š‘ ā‰¤2ā‹…(22š½āˆ’š‘˜2)āˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆ’āˆš22š½āˆ’š‘˜2šø.(3.15) Therefore,ā€–ā€–š‘ƒš½š‘‡š‘„(š¼āˆ’š‘ƒš½ā€–ā€–)š‘”š»š‘Ÿī€·ā‰¤21+š¶52ī€øī€·2š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆšāˆ’(1āˆ’š‘„)22š½āˆ’š‘˜2šøī€·2+22š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆ’āˆš22š½āˆ’š‘˜2šø.(3.16) Combining (3.11) and (3.16) with (3.10), we haveā€–ā€–(š‘‡š‘„āˆ’š‘‡š‘„,š½ā€–ā€–)š‘”š»š‘Ÿī€·š¶ā‰¤21+1+š¶52ī€øī€·2š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆšāˆ’(1āˆ’š‘„)22š½āˆ’š‘˜2šøī€·2+22š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆ’āˆš22š½āˆ’š‘˜2šø.(3.17) Then from (3.9) and (3.17) we finally arrive atā€–ā€–š‘‡š‘„š‘”āˆ’š‘‡š‘„,š½š‘”š‘šā€–ā€–š»š‘Ÿā‰¤ī‚€š¶5š‘’š‘„āˆš22(š½āˆ’1)āˆ’š‘˜2ī‚ī€·2+1š›æ+22š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆ’āˆš22š½āˆ’š‘˜2šøī€·š¶+21+1+š¶52ī€øī€·2š½āˆ’š‘˜2ī€øāˆ’(š‘ āˆ’š‘Ÿ)/2š‘’āˆšāˆ’(1āˆ’š‘„)22š½āˆ’š‘˜2šø.(3.18)

In order to show some stability estimates of the Hƶlder type for our method using (3.18), we use the following lemma which appeared in [24] for choosing a proper regularization parameter š½.

Lemma 3.4. Let the function š‘“(šœ†)āˆ¶[0,š‘Ž]ā†’ā„ be given by š‘“(šœ†)=šœ†š‘ī‚€1š‘‘lnšœ†ī‚āˆ’š‘(3.19) with a constant š‘āˆˆā„ and positive constants š‘Ž<1, š‘, and š‘‘. Then for the inverse function š‘“āˆ’1(šœ†), one has š‘“āˆ’1(šœ†)=šœ†1/š‘ī‚€š‘‘š‘1lnšœ†ī‚š‘/š‘(1+š‘œ(1))foršœ†āŸ¶0.(3.20)

Based on this lemma, we can choose the regularization parameter š½ by minimizing the right-hand side of (3.18).

Denoteš‘’āˆ’āˆš22š½āˆ’š‘˜2=šœ†āˆˆ(0,1),(3.21) and let š¶=š¶5/2(1+š¶1+š¶5) andš¶šœ†āˆ’š‘„š›æ=šœ†1āˆ’š‘„ī‚€1lnšœ†ī‚āˆ’(š‘ āˆ’š‘Ÿ)šø,(3.22) that is,š¶š›æšøī‚€1=šœ†lnšœ†ī‚āˆ’(š‘ āˆ’š‘Ÿ).(3.23) Then by Lemma 3.4 we obtain thatšœ†=š¶š›æšøī‚€1lnī‚š¶š›æ/šøš‘ āˆ’š‘Ÿ(1+š‘œ(1))forš¶š›æšø=āŸ¶0š¶š›æšøī‚€šølnī‚š¶š›æš‘ āˆ’š‘Ÿ(1+š‘œ(1))forš›æāŸ¶0.(3.24) Taking the principal part of šœ†, we getš½āˆ—=12log2ī‚µln2ī‚µšøī‚€šøš¶š›ælnī‚š¶š›æāˆ’(š‘ āˆ’š‘Ÿ)ī‚¶+š‘˜2ī‚¶,(3.25) due to (3.21). Now, summarizing above inference process, we obtain the main result of the present paper.

Theorem 3.5. For š‘ ā‰„š‘Ÿ, suppose that conditions (1.8) and (1.10) hold. If one takes š½āˆ—āˆ—=ī€ŗš½āˆ—ī€»,(3.26) where š½āˆ— was defined in (3.25), [š‘Ž] with square bracket denotes the largest integer less than or equal to š‘Žāˆˆš‘…. Then there holds the following stability estimate: ā€–ā€–š‘‡š‘„š‘”āˆ’š‘‡š‘„,š½š‘”š‘šā€–ā€–ā‰¤ī€·2ī€·š¶1+1+š¶5ī€øšøī€øš‘„ī€·š¶5š›æī€ø1āˆ’š‘„ī‚€šølnī‚š¶š›æāˆ’(š‘ āˆ’š‘Ÿ)š‘„Ɨī‚µī‚µ1+lnšø/š¶š›ælnšø/š¶š›æ+ln(lnšø/š¶š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶+ī‚µī‚µ1+2š¶lnšø/š¶š›ælnšø/š¶š›æ+ln(lnšø/š¶š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶š›æ=ī€·2ī€·š¶1+1+š¶5ī€øšøī€øš‘„ī€·š¶5š›æī€ø1āˆ’š‘„ī‚€šølnī‚š¶š›æāˆ’(š‘ āˆ’š‘Ÿ)š‘„(1+š‘œ(1)),(3.27) for š›æā†’0.

Remark 3.6. In general, the a priori bound šø and the coefficients š¶1-š¶5 and š¶ are not exactly known in practice. In this case, with š½ā‹†=ī‚ø12log2ī‚µln2ī‚µ1š›æī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶+š‘˜2ī‚¶ī‚¹,(3.28) it holds that ā€–ā€–š‘‡š‘„š‘”āˆ’š‘‡š‘„,š½š‘”š‘šā€–ā€–ā‰¤š›æ1āˆ’š‘„ī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)š‘„ī‚µš¶5ī€·š¶+21+1+š¶5ī€øšøī‚µln1/š›æln1/š›æ+ln(ln1/š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶+ī‚µš¶5ī‚µ+2šøln1/š›æln1/š›æ+ln(ln1/š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶š›æ=š›æ1āˆ’š‘„ī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)š‘„(1+š‘œ(1)),(3.29) for š›æā†’0.

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 [25]) Ī”š‘£(š‘„,š‘¦)+š‘˜2š‘£(š‘„,š‘¦)=0,š‘„āˆˆ(0,1),š‘¦āˆˆā„š‘›,š‘›ā‰„1,š‘£(0,š‘¦)=š‘”(š‘¦),š‘¦āˆˆā„š‘›,š‘£š‘„(0,š‘¦)=0,š‘¦āˆˆā„š‘›,(3.30) where Ī”=šœ•2/šœ•š‘„2+āˆ‘š‘›š‘–=1šœ•2/Ī”š‘¦2š‘– is the same as in (1.1).
It is easy to know that the exact solution of problem (3.30) is 1š‘£(š‘„,š‘¦)=(2šœ‹)š‘›/2ī€œā„š‘›š‘’š‘–šœ‰ā‹…š‘¦ī‚µš‘„ī”cosh||šœ‰||2+š‘˜2ī‚¶Ģ‚š‘”(šœ‰)š‘‘šœ‰.(3.31) Define an operator ī‚š‘‡š‘„āˆ¶š‘”(š‘¦)ā†¦š‘£(š‘„,š‘¦) such that ī‚æī‚š‘‡š‘„ī‚µš‘„ī”š‘”(šœ‰)=cosh||šœ‰||2+š‘˜2ī‚¶Ģ‚š‘”(šœ‰),0<š‘„ā‰¤1,(3.32) and the approximate solution is š‘£š›æš½=ī‚š‘‡š‘„,š½š‘”š‘š,(3.33) where š‘” and š‘”š‘š satisfy (1.8), and ī‚š‘‡š‘„,š½=š‘ƒš½ī‚š‘‡š‘„š‘ƒš½. If we select the regularization parameter š½ā€ =ī‚ø12log2ī‚µln2ī‚µ1š›æī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶āˆ’š‘˜2ī‚¶ī‚¹,(3.34) then there holds ā€–ā€–š‘£(š‘„,ā‹…)āˆ’š‘£š›æš½ā€–ā€–(š‘„,ā‹…)ā‰¤š›æ1āˆ’š‘„ī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)š‘„Ɨī‚µš¶5ī€·š¶+21+1+š¶5ī€øšøī‚µln1/š›æln1/š›æ+ln(ln1/š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶+ī‚µš¶5ī‚µ+2šøln1/š›æln1/š›æ+ln(ln1/š›æ)āˆ’(š‘ āˆ’š‘Ÿ)ī‚¶š‘ āˆ’š‘Ÿī‚¶š›æ=š›æ1āˆ’š‘„ī‚€1lnš›æī‚āˆ’(š‘ āˆ’š‘Ÿ)š‘„(1+(š‘œ(1))),(3.35) for š›æā†’0.

4. Numerical Aspect

4.1. Numerical Implementation

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

We consider the case when š‘›=2. Supposing that the sequence {š‘”(š‘¦1,š‘–,š‘¦2,š‘—)}š‘š‘–,š‘—=1 represents samples from the function š‘”(š‘¦1,š‘¦2) on an equidistant grid in the square [š‘Ž,š‘]2, and š‘ is even, then we add a random uniformly distributed perturbation to each data and obtain the perturbation dataš‘”š‘š=š‘”+šœ‡randn(size(š‘”)).(4.1) Then the total noise š›æ can be measured in the sense of root mean square error according toā€–ā€–š‘”š›æāˆ¶=š‘šā€–ā€–āˆ’š‘”š‘™2=ī„¶ī„µī„µī„µāŽ·1š‘2š‘ī“š‘š‘–=1ī“š‘—=1ī€·š‘”š‘šī€·š‘¦1,š‘–,š‘¦2,š‘—ī€øī€·š‘¦āˆ’š‘”1,š‘–,š‘¦2,š‘—ī€øī€ø2,(4.2) where ā€randn(ā‹…)ā€ is a normally distributed random variable with zero mean and unit standard deviation and šœ– dictates the level of noise. ā€randn(size(š‘”))ā€ returns an array of random entries that is the same size as š‘”.

For the function š‘”š‘š(š‘¦1,š‘¦2), we haveš‘¢š›æš½(š‘„,š‘¦)=š‘‡š‘„,š½š‘”š‘š=š‘ƒš½š‘‡š‘„š‘ƒš½š‘”š‘š.(4.3) Hence, by using it with š½ā‹† being given in (3.28), we can obtain the approximate solution.

We will use DMT as a short form of the ā€œdiscrete Meyer (wavelet) transform." Algorithms for discretely implementing the Meyer wavelet transform are described in [21]. These algorithms are based on the fast Fourier transform (FFT), and computing the DMT of a vector in ā„ requires š’Ŗ(š‘log22š‘) 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 šœ‡=10āˆ’3, š‘Ž=āˆ’5, š‘=5, and š‘=26.

Example 4.1. Take š‘›=2 and š‘”(š‘¦)=š‘’āˆ’š‘¦2āˆˆš’®(ā„2), where š‘¦=(š‘¦1,š‘¦2) and š’®(ā„2) denotes the Schwartz function space.
Since Ģ‚š‘”(šœ‰)āˆˆš’®(ā„2), šœ‰=(šœ‰1,šœ‰2) decays rapidly, and the formula (1.7) can be used to calculate š‘¢(š‘„,š‘¦) with exact data directly, that is, 1š‘¢(š‘„,š‘¦)=ī€œ2šœ‹ā„2š‘’š‘–(šœ‰1š‘¦1+šœ‰2š‘¦2)ī‚µš‘„ī”coshšœ‰21+šœ‰22āˆ’š‘˜2ī‚¶ī€·šœ‰Ģ‚š‘”1,šœ‰2ī€øš‘‘šœ‰1š‘‘šœ‰2.(4.4)

In Figure 1 we give the exact solution at š‘„=1, that is, š‘¢(1,š‘¦1,š‘¦2), and the reconstructed solution š‘¢š›æ(1,š‘¦1,š‘¦2) from the noisy data š‘”š‘š(š‘¦1,š‘¦2) without regularization. We see that š‘¢š›æ does not approximate the solution and some regularization procedure is necessary.

Letting š‘˜=1, the regularized solutions and the corresponding errors š‘¢āˆ’š‘¢š›æš½ā‹† defined by the regularization parameter š½ā‹†=3,4,5 are illustrated in Figure 2. We can see that in š‘‰3 the approximation is very poor since the frequencies are cut off excessively by the projection š‘ƒ3. If š½ā‹† is taken to be too large, the noise in the function š‘”š‘š is not damped enough by š‘ƒš½ā‹†, and thus the high frequencies of Ģ‚š‘”š‘š are so extremely magnified that they destroy the approximated solution. The approximation parameter š½ā‹†=4 seems to be the optimal choice for this example.

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

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

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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Copyright Ā© 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.


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