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Tikhonov regularization method for a

Cauchy problem for the Helmholtz

equation in multiple dimensions

∗

Qiu-Li Zhao, Chu-Li Fu

†

School of Mathematics and Statistics, Lanzhou University,

Lanzhou 730000, P. R. China

Abstract

This paper deals with a Cauchy problem for the Helmholtz equa-

tion in multiple dimensions. We want to obtain the solution u on the

boundary z = 0 from the given function g on the side z = 1. This

problem is an ill-posed problem: a small perturbation in the data g,

may cause dramatically large errors in the solution. Now the results

available in the literature are maily devoted to the case of two or three-

dimensional, and in most cases the stability theory and convergence

proofs on the boundary have not been generalized accordingly. The

present paper remedies this by a Tikhonov regularization method and

the stability estimate on the boundary has obtained. Numrical tests

of the method shows that the proposed method works well.

Key words: inverse problems,Helmholtze quation,Tikhonov regular-

ization,error estimate

∗

The project is supported by the NNSF of China (No. 10671085), the NSF of Gansu

Province of China (No. 3ZS051-A25-015) and the Fundamental Research Fund for Physics

and Mathematic of Lanzhou University (No. Lzu05-05).

†

Corresponding author. E-mail: fuchuli@lzu.edu.cn

http://www.paper.edu.cn

1 Introduction

In many physical and engineering applications, it is sometimes necessary

to reconstruct the radiation ﬁled from experimental data on a part of the

boundary. This so-called inverse scattering problems leads to the Cauchy

problem for Helmholtz equation[1,3,4,5,6,7,9]. After been studied by many

authors a number of numerical methods for stabilizing this problem are de-

veloped. However,these numerical methods are short of stability theory and

convergence proofs. In [7]the applications for a model of Helmholtz equation

are introduced, a method by cutting oﬀ high frequency directly is applied for

solving a Cauchy problem for Helmholtz equation, some error estimates are

also obtained. Recently in[9], some spectra methods and a revised Tikhonov

regularization method which are diﬀerent from the method used in [7] are

discussed, the Holder-type stability estimate for z ∈ (0, d) is aslo obtained.

However their method is woking only for the Hilb ert space L

. Note that

the results available in the literatures are mainly devoted to two or three-

dimensional cases and the convergence proofs on the boundary couldn’t been

found in most papers.

The aim of this paper is to consider a Cauchy problem for the Helmholtz

equation in multiple dimensions, especially give the convergence proofs on

the boundary z = 0 by using the Tikhonov regularization method. It is

organized as follows: in Section 2, the formulation of solution of problem (1.1)

is given and the ill-posedness is discussed; Section 3 is devoted to proving

some auxiliary lemmas; in section 4, a Tikhonov regularization method with

error estimate is provided; the last section present some numerical results.

We consider a Cauchy problem for the Helmholtz equation in multiple

dimensions as follows:

∆u(y, z) + k

u(y, z) = 0, y ∈ R

, z ∈ [0, 1]

u( y, 1) = g(y), y ∈ R

( y, 1) = 0, y ∈ R

(1.1)

where ∆ is the Laplace operator in n+1(n ∈ N) dimensions, g(·) ∈ L

), u( ·, z) ∈

n+1

).Here we want to obtain the solution u( y, 0) form the Cauchy

data u( y, 1) := g(y), y ∈ R

. In the next section we analyze the ill-posedness

of problems(1.1).

http://www.paper.edu.cn

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