A Wavelet Method for the Cauchy Problem for the Helmholtz Equation
Fang-Fang Dou1and Chu-Li Fu2
Academic Editor: A. Bellouquid, L. Marin
Received16 Aug 2011
Accepted28 Sept 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 Earthโs 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 โstripโ as follows:
where is an dimensional Laplace operator. We want to determine the solution for from the data . 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. [10โ14].
Let be the Schwartz space over , and let be its dual (the space of tempered distributions). Let denote the Fourier transform of function defined by
while the Fourier transform of a tempered distribution is defined by
In this paper, we will consider functions depending on the variables , .
For , the Sobolev space consists of all tempered distributions , for which is a function in . The norm on this space is given by
We assume there exists a unique solution of problem (1.1), which satisfies the problem in the classical sense and , . Applying the Fourier transform technique to problem (1.1) with respect to the variable yields the following problem in the frequency space:
It is easy to obtain the solution of problem (1.5) (if exists) has the form
or equivalently, the solution of problem (1.1) has the representation
Since increases rapidly with exponential order as , the Fourier transform of the exact data must decay rapidly. However, in practice, the data at is often obtained on the basis of reading of physical instrument which is denoted by . We assume that and satisfy
Since belong to for , should not be positive. A small perturbation in the data may cause a dramatically large error in the solution for . 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 know
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 by considering
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 be Meyerโs orthonormal scaling function in dimensions. This function is constructed from the one-dimensional scaling functions in the following way. Let and be the Meyer scaling and wavelet function in one dimension defined by their Fourier transform in [20] which satisfy
It can be proved (cf. [20]) that the set of functions
is an orthonormal basis of . Consequently, the MRA of Meyer is generated by
For the construction of an -dimensional MRA, we take tensor products of the spaces (see [21, 22]). Then the scaling function is given by
and any basis function in can be written in the form
where , and for any , stands for or . Hence we obtain from (2.1) that
The orthogonal projection on the space is defined by
while denotes the orthogonal projection of a function on the wavelet space with . (In many contexts one will find more than one detailed space , that is, . Here, the space is simply defined as the orthogonal complement of in ).
Let
Setting , together with (2.6), it follows for that
We introduce the operator which is defined by the equation
where denotes the characteristic function of the cube . From (2.7) it follows that any basis function in , , satisfies
and we obtain
And it follows for that
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 be an m-regular MRA, and let be such that . Then for each function and , the following inequality holds:
Lemma 3.2 (see [19]). Let be Meyerโs (tensor-) MRA, and suppose , . Then for all , one has
Define an operator by (1.6), that is,
or equivalently,
Then we have
Theorem 3.3. Let be Meyerโs MRA and suppose and which satisfies , . Then for all , one has
Proof. For , by definition (1.4) and formula (3.4), from Lemma 3.2, we have
Since the Cauchy data are given inexactly by , we need a stable algorithm to approximate the solution of (1.1). Our method is as follows. Consider the operator
and show that it approximates in a stable way for an appropriate choice for depending on and . By the triangle inequality we know
From (1.8) and Theorem 3.3, the second term on the right-hand side of (3.8) satisfies
For the first one we have
By Lemma 3.1, (1.9), (1.10), (2.14), and (3.4), we get
On the other hand, due to (2.10), we know
We estimate the two parts at the right-hand side of (3.12) separately. For we have
Now we turn to . There holds
since . Furthermore, from (2.14), it follows that
Therefore,
Combining (3.11) and (3.16) with (3.10), we have
Then from (3.9) and (3.17) we finally arrive at
In order to show some stability estimates of the Hรถlder type for our method using (3.18), we use the following lemma which appeared in [24] for choosing a proper regularization parameter .
Lemma 3.4. Let the function be given by
with a constant and positive constants , , and . Then for the inverse function , one has
Based on this lemma, we can choose the regularization parameter by minimizing the right-hand side of (3.18).
Denote
and let and
that is,
Then by Lemma 3.4 we obtain that
Taking the principal part of , we get
due to (3.21). Now, summarizing above inference process, we obtain the main result of the present paper.
Theorem 3.5. For , suppose that conditions (1.8) and (1.10) hold. If one takes
where was defined in (3.25), with square bracket denotes the largest integer less than or equal to . Then there holds the following stability estimate:
for .
Remark 3.6. In general, the a priori bound and the coefficients - and are not exactly known in practice. In this case, with
it holds that
for .
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])
where is the same as in (1.1). It is easy to know that the exact solution of problem (3.30) is
Define an operator such that
and the approximate solution is
where and satisfy (1.8), and . If we select the regularization parameter
then there holds
for .
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 . Supposing that the sequence represents samples from the function on an equidistant grid in the square , and is even, then we add a random uniformly distributed perturbation to each data and obtain the perturbation data
Then the total noise can be measured in the sense of root mean square error according to
where is a normally distributed random variable with zero mean and unit standard deviation and dictates the level of noise. returns an array of random entries that is the same size as .
For the function , we have
Hence, by using it with being given in (3.28), we can obtain the approximate solution.
We will use DMT as a short form of the โdiscrete 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 requires 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 , , , and .
Example 4.1. Take and , where and denotes the Schwartz function space. Since , decays rapidly, and the formula (1.7) can be used to calculate with exact data directly, that is,
In Figure 1 we give the exact solution at , that is, , and the reconstructed solution from the noisy data without regularization. We see that does not approximate the solution and some regularization procedure is necessary.
(a)
(b)
Letting , the regularized solutions and the corresponding errors defined by the regularization parameter are illustrated in Figure 2. We can see that in the approximation is very poor since the frequencies are cut off excessively by the projection . If is taken to be too large, the noise in the function is not damped enough by , and thus the high frequencies of are so extremely magnified that they destroy the approximated solution. The approximation parameter seems to be the optimal choice for this example.
In Figure 3 we display the exact solution, its approximation, and corresponding errors for and 100, respectively. We see that the proposed method is useful for different wave number .
Figure 4 shows that the proposed method for the Cauchy problem for the modified Helmholtz equation is also effective.
(a)
(b)
(c)
(d)
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).
References
N. Bleistein, J. K. Cohen, and J. W. Stockwell, Jr., Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, vol. 13, Springer, New York, NY, USA, 2001.
A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics, IEEE Press, New York, NY, USA, 1998.
A. V. Kononov, C. D. Riyanti, S. W. de Leeuw, C. W. Oosterlee, and C. Vuik, โNumerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation,โ Computing and Visualization in Science, vol. 11, no. 3, pp. 139โ146, 2008.
T. Regiลska and K. Regiลski, โApproximate solution of a Cauchy problem for the Helmholtz equation,โ Inverse Problems, vol. 22, no. 3, pp. 975โ989, 2006.
J. Hadamard, Lectures on Cauchy Problem in Linear Partial Differential Equations, University Press, London, UK, 1923.
T. DeLillo, V. Isakov, N. Valdivia, and L. Wang, โThe detection of the source of acoustical noise in two dimensions,โ SIAM Journal on Applied Mathematics, vol. 61, no. 6, pp. 2104โ2121, 2001.
T. DeLillo, V. Isakov, N. Valdivia, and L. Wang, โThe detection of surface vibrations from interior acoustical pressure,โ Inverse Problems, vol. 19, no. 3, pp. 507โ524, 2003.
B. T. Jin and Y. Zheng, โA meshless method for some inverse problems associated with the Helmholtz equation,โ Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 19โ22, pp. 2270โ2288, 2006.
B. T. Johansson and L. Marin, โRelaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation,โ CMC. Computers, Materials, & Continua, vol. 13, no. 2, pp. 153โ189, 2010.
L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, and X. Wen, โAn alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation,โ Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 5-6, pp. 709โ722, 2003.
L. Marin and D. Lesnic, โThe method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations,โ Computers & Structures, vol. 83, no. 4-5, pp. 267โ278, 2005.
L. Marin, โA meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations,โ Engineering Analysis with Boundary Elements, vol. 34, no. 3, pp. 274โ288, 2010.
L. Marin, โBoundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations,โ Computational Mechanics, vol. 44, no. 2, pp. 205โ219, 2009.
L. Marin, โAn alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation,โ Computational Mechanics, vol. 45, no. 6, pp. 665โ677, 2010.
D. N. Hร o, A. Schneiderx , and H. J. Reinhardt, โRegularization of a noncharacteristic Cauchy problem for a parabolic equation,โ Inverse Problems, vol. 11, pp. 124โ1263, 1995.
Y. Knosowski, E. von Lieres, and A. Schneider, โRegularization of a non-characteristic Cauchy problem for a parabolic equation in multiple dimensions,โ Inverse Problems, vol. 15, no. 3, pp. 731โ743, 1999.
E. D. Kolaczyk, Wavelet methods for the inversion of certain homogeneous linear operators in the presence of noisy data, Ph.D. thesis, Stanford University, Stanford, Calif, USA, 1994.
Y. Meyer, Wavelets and Operators, vol. 37, Cambridge University Press, Cambridge, UK, 1992.
A. Schneider, Wavelet-based mollication methods for some ill-posed problems, Dissertation, Universitรคt Siegen, 1996.
U. Tautenhahn, โOptimality for ill-posed problems under general source conditions,โ Numerical Functional Analysis and Optimization, vol. 19, no. 3-4, pp. 377โ398, 1998.
H. W. Chang, J. F. Huang, and T. J. Leiterman, โAn adaptive fast solver for the modified Helmholtz equation in two dimensions,โ Journal of Computational Physics, vol. 211, pp. 616โ637, 2003.
Copyright ยฉ 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.