A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation

Volume 2010, Article ID 212056,13pages doi:10.1155/2010/212056

Research Article

A Spectral Regularization Method for a Cauchy

Problem of the Modified Helmholtz Equation

Ailin Qian, Jianfeng Mao, and Lianghua Liu

School of Mathematics and Statistics, Xianning College, Xianning, Hubei 437100, China

Correspondence should be addressed to Ailin Qian,[email protected]

Received 15 December 2009; Accepted 9 May 2010

Academic Editor: Salim Messaoudi

Copyrightq2010 Ailin Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the Cauchy problem for a modified Helmholtz equation, where the Cauchy data is given atx 1 and the solution is sought in the interval 0 < x < 1. A spectral method together with choice of regularization parameter is presented and error estimate is obtained. Combining the method of lines, we formulate regularized solutions which are stably convergent to the exact solutions.

1. Introduction

The Cauchy problem for Helmholtz equation arises from inverse scattering problems. Specific backgrounds can be seen in the existing literature; we can refer to 1–6 and so forth. A number of numerical methods for stabilizing this problem are developed. Several boundary element methods combined with iterative, conjugate gradient, Tikhonov regularization, and singular value decomposition methods are compared in 6. Cauchy problem for elliptic equations is well known to be severely ill-posed7; that is, the solution does not depend continuously on the boundary data, and small errors in the boundary data can amplify the numerical solution infinitely; hence it is impossible to solve Cauchy problem of Helmholtz equation by using classical numerical methods and it requires special techniques, for exam-ple, regularization methods. Although theoretical concepts and computational implementa-tion related to the Cauchy problem of Helmholtz equaimplementa-tion have been discussed by many authors 8–11, there are many open problems deserved to be solved. For example, many authors have considered the following Cauchy problem of the Helmholtz equation8–11:

uxxuyy−k2u

x, y0, 0< x <1, −∞< y <∞,

u1, ygy, −∞< y <∞,

ux

1, y0, −∞< y <∞.

However, the boundary conditionux1, y 0 is very strict. If the boundary condition

is replaced byux1, y hy, wherehyis a function ofy, then their methods cannot be

applied easily. Therefore, in this paper, we consider the following Cauchy problem for the Helmholtz equation:

uxxuyy−k2u

x, y0, 0< x <1, −∞< y <∞,

u1, ygy, −∞< y <∞,

ux

1, yhy, −∞< y <∞.

1.2

We want to seek the solution in the interval0,1from the Cauchy data pairsg, h

located at x 1. Of course, since g and h are assumed to be measured, there must be measurement errors, and we would actually have noisy data function gδ, hδ ∈ L2R, for

which the measurement errorsh−hδand g−gδare small. Here and in the following

sections, · denotes theL2_{Rnorm. Thus1.2is a noncharacteristic Cauchy problem with} appropriate Cauchy datau, uxgiven on the linex1.

However, the ill-posedness is caused by high frequency. By introducing a “cutoff” frequency we can obtain a well-posed problem. This method has been used for solving inverse heat conduction problem12, and sideways heat equation13, sideways parabolic equation 14. An error estimate for the proposed method can be found inSection 2. The implementation of the numerical method is explained.

2. Regularization and Error Estimate

First, letube the Fourier transform ofu:

ux, ξ √1

2π

_∞

−∞u

x, ye−iξydy, ξ∈R. 2.1

Taking Fourier transformation for1.2, we have a family of problems parameterized byξ:

uxxx, ξ−ξ2ux, ξ−k2ux, ξ 0, 0< x <1, ξ∈R,

u1, ξ gξ, ξ∈R,

ux1, ξ hξ, ξ∈R.

2.2

The solution can easily be verified to be

ux, ξ −hξ ξ2_k2sinh

1−x

ξ2_k2 _gξcosh

1−x

Following the idea of Fourier method, we consider2.2only for|ξ|< ξcby cutting off

high frequency and define a regularized solution:

ucx, ξ χcξux, ξ, 2.4

whereχc is the characteristic function of the interval−ξc, ξc. The solutionucx, ycan be

found by using the inverse Fourier transform. Define the regularized solution with measured data gδ, hδ byuδc. The difference between the exact solution ux, yand the regularized

solutionuδ

cx, ycan be divided into two parts:

ux,·−uδ

cx,· ≤ u−ucuc−uδcRaRb, 2.5

where

R2

a

|ξ|>ξc

|ux, ξ|2dξ, R2

b

ξc

−ξc

|uc−uδc|

2

dξ. 2.6

We rewrite2.2as a system of ordinary differential equations:

ξ2_k2_u

ux

x

0 ξ2_k2

ξ2_k2 ₀

ξ2_k2_u

ux

, 0≤x≤1. 2.7

LettingUx ξ2_k2_{u ux}T_{, we can rewrite2.7as}

Uxx AUx, 0≤x≤1, U1

ξ2_k2_g

h

, 2.8

where the matrixA is the one in2.7. The reason for using ξ2_k2_{u instead ofu in the} definition ofUxis that with this choice, the matrixAis normal and hence diagonalized by a unitary matrix. The eigenvalues ofAareλ1,2±

ξ2_k2_{. Thus we can factorizeAas}

AXΛXH, Λ

ξ2_k2 ₀ 0 −ξ2_k2

, 2.9

whereXis a unitary matrix,XH_{X I; it follows that the solution of the system2.8can be}

written as

Ux XeΛx−1_XH_{U1. 2.10}

SinceXis unitary,X1 and therefore

where · ₂ denotes both the Euclidean norm in the complex vector space C2 _{and the} subordinate matrix norm, rememberingU ξ2_k2_{u ux}T_{in2.11, we obtain}

ξ2_k2_u 2

|ux|2 ≤e2

√

ξ2_k2_1−x

ξ2_k2_{u1, ξ} 2

|ux1, ξ|2

. 2.12

This inequality is valid for allξ∈Rand we can integrate overξand use the Parseval theorem to obtain estimates for u and ux in the L2R-norm. First we will prove a bound on the

difference between any two regularized solutions2.4. We have errors in the measuredgy

andhy. These two cases are treated separately.

Lemma 2.1. Assume that one has two regularized solutions u1 and u2 defined by 2.4, with the

Cauchy datag, h1andg, h2. Then

u1x,·−u2x,· ≤

e√ξ2

ck21−x−₁

ξc2k2

h1−h2. 2.13

Proof. The functionw u1−u2 satisfies the differential equation; thusw solves2.2, with

initial data given by

w1,· 0, wx1,·

h1−h2

χc, 2.14

and thuswsatisfies inequality2.12. We have

wxx, ξ2≤e2

√

ξ2_k2_1−x

|wx1, ξ|2≤e2

√

ξ2

ck21−x|_h 1−h2|

2

. 2.15

If 0≤z≤1, then

wz, ξ −

₁

z

wxx, ξdxw1, ξ − ₁

z

wxx, ξdx 2.16

holds. By inserting2.15into2.16we get

|wz, ξ|

1

z

wxx, ξdx

≤

1

z

|wx|dx≤ 1

z e

√

ξ2_ck2_1−x

|wx1, ξ|dx

e

√

ξ2

ck21−z−₁

ξ2

ck2

· |wx1, ξ|.

If we insertwu1−u2and integrate overξ, then we obtain

_∞

−∞u1−u2

2_dξ≤ ⎛ ⎜

⎝e

√

ξ2

ck21−x−₁

ξ2

ck2

⎞ ⎟ ⎠ 2

·

_∞

−∞

h1ξ−h2ξ 2

dξ. 2.18

Thus2.13holds.

Lemma 2.2. Assume that one has two regularized solutions u1 and u2 defined by 2.4, with the

Cauchy datag1, handg2, h. Then

u1x,·−u2x,· ≤e √

ξ2

ck21−x_g

1−g2. 2.19

Proof. The functionwu1−u2satisfies the differential equation, with initial data given by

w1, ξ g1−g2

χc, wx1, ξ 0, 2.20

and thus inequality2.12holds. It follows that

ξ2_k2_{wx, ξ} 2

≤e2

√

ξ2_k2_1−x

|ξw1, ξ|2, 2.21

and thus, forξ2_k2_/0,

wx, ξ2 ≤e2√ξ2k21−x|w1, ξ|2. 2.22

By integrating over the interval−ξc, ξc, we get

ξc

−ξc

|wx, ξ|2dξ≤

ξc

−ξc

e2

√

ξ2_k2_1−x

|w1, ξ|2dξ≤e2

√

ξ2

ck21−x ξc

−ξc

|w1, ξ|2dξ. 2.23

Sincew u1−u2 is equal to zero outside the interval−ξc, ξc, we can extend the integrals.

Insertingwu1−u2,we get

_∞

−∞u1−u2

2_dξ≤e2√ξ2_ck2_1−x

·

_∞

−∞|g1−g2|

2_{dξ. 2.24}

This is precisely2.19.

Lemma 2.3stability. Assume thatucis the regularized solution2.4, with exact datag, h,and uδ

cis the regularized solution with noisy datagδ, hδ; then

uc−uδc ≤e

√

ξ2

ck21−x ⎛ ⎜

⎝g−gδ h−hδ

ξ2

ck2

⎞ ⎟

⎠. 2.25

Proof. Letu1be a regularized solution defined by2.4, with datag, hδ. Then byLemma 2.1,

uc−u1 ≤ e√ξ2

ck21−x−₁

ξ2ck2

h−hδ ≤

e√ξ2_ck2_1−x

ξ2ck2

h−hδ. 2.26

UsingLemma 2.2, we have

u1−uδc ≤e

√

ξ2_ck2_1−x

g−gδ. 2.27

By the triangle inequality,

uc−uδc ≤ uc−u1u1−uδc ≤e

√

ξ2_ck2_1−x

⎛ ⎜

⎝g−gδh−hδ ξ2

ck2

⎞ ⎟

⎠. 2.28

This completes the proof.

By Lemma 2.3the regularized solution depends continuously on the data. Next we derive a bound on the truncation error when we neglect frequencies|ξ| ≥ξc in2.3. So far

we have not used any information aboutuforx >1; we, assume that the Helmholtz equation

uxxuyy−k2u0 is valid in a large interval 0< x < L, L >1. By imposing a priori bounds on

the solution atx0 and atx > 1, we obtain an estimate of the difference between the exact solutionuand a regularized solutionucwith “cutoff” levelξc. This is a convergence result

in the sense thatuc → uasξc → ∞for the case of exact datag, h. The following estimate

holds.

Lemma 2.4 convergence. Suppose that u is the solution of the problem 2.2, and that the

Helmholtz equation is valid for0 < x < L. Then the difference betweenuand a regularized solution

uccan be estimated:

ux,·−ucx,· ≤C

e−x

√

ξ2

ck2_u0,·_ex−L √

ξ2

ck2_uL,·

, 2.29

where the constantCis defined by

C sup 0≤x≤L,

ξ /0

1−e−2x√ξ2_k2

Proof. The solution of2.2can be written in the floowing form:

uAex

√

ξ2_k2

Be−x

√

ξ2_k2

, 2.31

whereAandBcan be determined from the boundary conditions:

1 1

eL√ξ2_k2

e−L√ξ2_k2

A

B

u0, ξ

uL, ξ . 2.32

Solving forAandBwe find that the solution can be written:

ux, ξ ϕL−x, ξu0, ξ ϕx, ξuL, ξ, 2.33

where

ϕx, ξ e

x√ξ2_k2₋

e−x√ξ2k2 eL√ξ2_k2₋

e−L√ξ2_k2 e

x−L√ξ2_k21−e−2x

√

ξ2_k2

1−e−2L√ξ2_k2. 2.34

We make the observation that

_{ϕx, ξ≤Ce}x−L√ξ2_k2

2.35

and that

ϕL−x, ξ≤Ce−x√ξ2_k2

. 2.36

Using the expression2.33and the triangle inequality, we get

u−uc ≤ ϕL−x,·u0,·−uc0,·ϕx,·uL,·−ucL,·. 2.37

The first term on the right-hand side satisfies

ϕL−x,·u0,·−uc0,·2 ≤

|ξ|>ξc

C2e−2x

√

ξ2_k2

|u0, ξ|2dξ

≤C2e−2x

√

ξ2_ck2

|ξ|>ξc

|u0, ξ|2dξ

≤C2e−2x

√

ξ2_ck2

u0, ξ2.

Similarly, the second term can be estimated:

ϕx,·uL,·−ucL,·2≤C2e−2L−x

√

ξ2_ck2

· uL, ξ2. 2.39

By combining these two expressions, we have

ux,·−ucx,· ≤C

e−x

√

ξ2

ck2_u0,·_e−L−x √

ξ2

ck2_uL,·

. 2.40

Thus the proof is complete.

Remark 2.5. The constantCis well defined, but its value has to be estimated. From a numerical

computation we conclude thatC <2.

Remark 2.6. When solving2.2numerically we need Cauchy datau, ux gδ, hδ, along

the line x 1. The most natural way to obtain this is to use two thermocouples, located at x 1, andx L, and computehδ by solving a well-posed problem for the Helmholtz

equation in the interval1, L. Hence it is natural to assume knowledge aboutuat a second point.

Let us summarize what we have so far. The constantCinLemma 2.4is unchanged. The propagated data error is estimated usingLemma 2.3:

Rbuc−uδc ≤e

√

ξ2

ck21−x ⎛ ⎜

⎝g−gδ h−hδ ξ2

ck2

⎞ ⎟

⎠, 2.41

and the truncation error is estimated usingLemma 2.4:

Rau−uc ≤C

e−x

√

ξ2

ck2_u0,·_ex−L √

ξ2

ck2_uL,·

. 2.42

These two results can be combined into an error estimate for the spectral method. This is demonstrated in two examples.

Example 2.7. Suppose that ux,· ≤ E, and that we have an estimate of the noise level,

g−gδh−hδ ≤ δ. IfL ≥3 and if we chooseξc

logE/δ2−k2_{, with this choice,} then by expression2.42,

Ra≤Ee−x

√

ξ2

ck2_C

1e2x−L√ξ2 ck2

where we have assume that 0< x <1 and used the boundC <2. By expression2.41,

Rb≤e

√

ξ2

ck21−x_δ≤_E1−x_δx_. 2.44

Thus we obtain an error estimate:

ux,·−uδ_cx,· ≤RaRb≤3E1−xδx. 2.45

Note that, under these assumptions,ξc

logE/δ2−k2_{can be used as a rule for selecting} the regularization parameter.

Example 2.8. Suppose that the Helmholtz equation is valid in the interval 0 ≤ x ≤ 5 and

that we have a priori boundsu0,· ≤ 1 andu5,· ≤0.5. Furthermore, we assume that the measured data satisfies g −gδ ≤ 10−3 and h−hδ ≤ 0.5·10−2. Then we have the

estimates:

Rb uc−uδc ≤e1−x

√

ξ2_ck2

⎛ ⎜

⎝10−30.5· 10− 2

ξ2

ck2

⎞ ⎟ ⎠,

Rau−uc ≤2

e−x

√

ξ2

ck2·_1ex−5 √

ξ2

ck2·_0.5

.

2.46

By balancing these two components Ra and Rb, we can find a suitable value for the

regularization parameter.

3. Numerical Implementation

In this section, we use the method of Lines. Let

UUx

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

U0x

U1x

U2x .. .

Unx ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ≈ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Ux, y0

Ux, y1

Ux, y2

.. .

Ux, yn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , yi i

n, i0,1,2, . . . , n,

−4 −3 −2 −1 0 1 2 3 4 −0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

y Approximation Exact solution u 0 . 4 ,y and its appr oximation

Figure 1:x0.4, δ1∗10−5, ξc11.35.

and then we can obtain

Ux U x 0 D I 0 Ux U ,

Ux1

U1

H G , 3.2 where H ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ux

1, y0

Ux

1, y1

Ux

1, y2

.. .

Ux

1, yn ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , G ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

U1, y0

U1, y1

U1, y2

.. .

−4 −3 −2 −1 0 1 2 3 4 −0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

y

Approximation Exact solution

u

0

,y

and

its

appr

oximation

Figure 2:x0, δ1∗10−5, ξc11.35.

−4 −3 −2 −1 0 1 2 3 4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

y

Approximation Exact solution

u

0

.

4

,y

and

its

appr

oximation

Figure 3:x0.4, δ1∗10−2, ξc4.45.

The matrixDapproximates the second-order derivativeuyy by spectral cutoffmethodsee

2. This ordinary differential equation system can easily be solved by various numerical methods.

Example 3.1. It is easy to verify that the functionux, y sinxcosh√1k2_y/5k2_{is the}

exact solution of1.2. Now we need to seek the solutionux, y sinxcosh√1k2_y/5

k2_{, where 0≤x <1 from the Cauchy data pairsu1, y sin 1 cosh}√_1k2_y/5k2_and

ux1, y cos 1 cosh

√

−4 −3 −2 −1 0 1 2 3 4 −0.2

0 0.2 0.4 0.6 0.8 1 1.2

y

Approximation Exact solution

u

0

,y

and

its

appr

oximation

Figure 4:x0, δ1∗10−2, ξc4.45.

In our numerical experiment, we give the comparison between the exact solution of problem2.2and its approximation by spectral cutoffmethod for different noise levelsδand different locationsx. Please see Figures 1,2,3, and4. In the experiment, the regularization parameter is chosen according to the Remark ofRemark 2.5, where we takeE1. From the above results, it is easy to see that the numerical effect of the spectral method works well. Moreover, we can also see that the lower the noise levelδis, the better the approximate effect is; the closer to the boundaryx1 the location is, the better the approximate effect is. These accord with the theory inSection 2.

Acknowledgments

The author wants to express his thanks to the referee for many valuable comments. This work is supported by the Educational Commission of Hubei Province of ChinaQ20102804, T201009.

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