A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

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 Earth’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 “strip” 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. [10–14].

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 Meyer’s 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 Meyer’s (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 Meyer’s 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 Hölder 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 “discrete 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,