Open Access

Solving the Axisymmetric Inverse Heat Conduction Problem by a Wavelet Dual Least Squares Method

Boundary Value Problems20092009:260941

DOI: 10.1155/2009/260941

Received: 17 August 2008

Accepted: 10 March 2009

Published: 17 March 2009

Abstract

We consider an axisymmetric inverse heat conduction problem of determining the surface temperature from a fixed location inside a cylinder. This problem is ill-posed; the solution (if it exists) does not depend continuously on the data. A special project method—dual least squares method generated by the family of Shannon wavelet is applied to formulate regularized solution. Meanwhile, an order optimal error estimate between the approximate solution and exact solution is proved.

1. Introduction

Inverse heat conduction problems (IHCP) have become an interesting subject recently, and many regularization methods have been developed for the analysis of IHCP [113]. These methods include Tikhonov method [1, 2], mollification method [3, 4], optimal filtering method [5], lines method [6], wavelet and wavelet-Galerkin method [711], modified Tikhonov method [12] and "optimal approximations" [13], and so forth. However, most analytical and numerical methods were only used to dealing with IHCP in semiunbounded region. Some works of numerical methods were presented for IHCP in bounded domain [1419].

Chen et al. [14] applied the hybrid numerical algorithm of Laplace transform technique to the IHCP in a rectangular plate. Busby and Trujillo [15] used the dynamic programming method to investigate the IHCP in a slab. Alifanov and Kerov [16] and Louahlia-Gualous et al. [17] researched IHCP in a cylinder. However to the authors' knowledge, most of them did not give any stability theory and convergence proofs.

In this paper, we will treat with a special IHCP whose physical model consists of an infinitely long cylinder of radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq1_HTML.gif . It is considered axisymmetric and a thermocouple (measurement equipment of temperature) is installed inside the cylinder (at the radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq3_HTML.gif ). The correspondingly mathematical model of our problem can be described by the following axisymmetric heat conduction problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ1_HTML.gif
(1.1)

where the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq5_HTML.gif belong to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq6_HTML.gif for every fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq8_HTML.gif is the radial coordinate, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq9_HTML.gif denotes the temperature history at one fixed radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq10_HTML.gif of cylinder. We want to recover https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq11_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq12_HTML.gif . This problem is ill-posed problem; a small perturbation in the data may cause dramatically large errors in the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq13_HTML.gif (The details can be seen in Section 2).

To the authors' knowledge, up to now, there is no regularization theory with error estimate for problem (1.1) in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq14_HTML.gif . The major objective of this paper is to do the theoretic stability and convergence estimates for problem (1.1).

Xiong and Fu [11] and Regińska [20] solved the sideways heat equation in semi-unbounded region by applying the wavelet dual least squares method, which is based on the family of Meyer wavelet. In this paper, we will apply a wavelet dual least squares method generated by the family of Shannon wavelet to problem (1.1) in bounded domain for determining surface temperature. According to the optimality results of general regularization theory, we conclude that our error estimate on surface temperature is order optimal.

2. Formulation of Solution of Problem (1.1)

As we consider problem (1.1) in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq15_HTML.gif with respect to variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq16_HTML.gif , we extend https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq17_HTML.gif , and other functions of variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq18_HTML.gif appearing in the paper to be zero for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq19_HTML.gif . Throughout the paper, we assume that for the exact https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq20_HTML.gif the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq21_HTML.gif exists and satisfies an apriori bound
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ2_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq22_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ3_HTML.gif
(2.2)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq23_HTML.gif is measured by the thermocouple, there will be measurement errors, and we would actually have as data some function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq24_HTML.gif , for which
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ4_HTML.gif
(2.3)
where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq25_HTML.gif represents a bound on the measurement error, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq26_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq27_HTML.gif norm and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ5_HTML.gif
(2.4)
is the Fourier transform of function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq28_HTML.gif . The problem (1.1) can be formulated, in frequency space, as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ6_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ7_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ8_HTML.gif
(2.7)

Then we have the following lemma.

Lemma 2.1.

Problem (2.5)–(2.7) has the solution given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ9_HTML.gif
(2.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq29_HTML.gif denotes modified spherical Bessel function which given by [21]
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ10_HTML.gif
(2.9)

Proof.

Due to [21], we can solve (2.5), in the frequency domain, to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ11_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq30_HTML.gif denotes also modified spherical Bessel function which is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ12_HTML.gif
(2.11)
Combining https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq31_HTML.gif with condition(2.7), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq32_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ13_HTML.gif
(2.12)
According to [21], there holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ14_HTML.gif
(2.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq33_HTML.gif , both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq35_HTML.gif denote the Kelvin functions. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq36_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ15_HTML.gif
(2.14)
Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq37_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ16_HTML.gif
(2.15)
Solving the systems (2.6) and (2.12) using (2.15) we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ17_HTML.gif
(2.16)

Substitution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq38_HTML.gif in (2.16) into (2.12), we obtain (2.8).

Applying an inverse Fourier transform to (2.8), problem (1.1) has the solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ18_HTML.gif
(2.17)

In order to obtain ill-posedness of problem (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq39_HTML.gif , we need the following lemma.

Lemma 2.2.

If function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq40_HTML.gif satisfies (2.15), then there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq41_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq42_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ19_HTML.gif
(2.18)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ20_HTML.gif
(2.19)

Proof.

First, due to [21] and (2.15), we have, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq44_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ21_HTML.gif
(2.20)
then there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq45_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq46_HTML.gif large enough, say https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq47_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ22_HTML.gif
(2.21)
From these we know that there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq49_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq51_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ23_HTML.gif
(2.22)
Then, since function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq52_HTML.gif is continuous in the closed region https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq53_HTML.gif . Threrfore, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq55_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq57_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ24_HTML.gif
(2.23)

Finally, combining inequalities (2.22) with (2.23), we can see that there exist others constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq59_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq60_HTML.gif , inequalities (2.18) are valid. Similarly, we obtain inequalities (2.19).

In order to formulate problem (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq61_HTML.gif in terms of an operator equation in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq62_HTML.gif , we define an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq63_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ25_HTML.gif
(2.24)
From (2.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ26_HTML.gif
(2.25)
Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq64_HTML.gif , and we can see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq65_HTML.gif is a multiplication operator:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ27_HTML.gif
(2.26)

From (2.26), we can prove the following lemma.

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq66_HTML.gif be the adjoint to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq67_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq68_HTML.gif corresponds to the following problem where the left-hand side https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq69_HTML.gif of problem (1.1) is replaced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq70_HTML.gif , says
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ28_HTML.gif
(2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ29_HTML.gif
(2.28)

Proof.

Via the the following relations, combining with (2.26),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ30_HTML.gif
(2.29)
we can get the adjoint operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq71_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq72_HTML.gif in frequency domain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ31_HTML.gif
(2.30)
On the other hand, the problem (2.27) can be formulated, in frequency space, as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ32_HTML.gif
(2.31)
Taking the conjugate operator for problem (2.5)–(2.7), we realize that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq73_HTML.gif . Therefore, by Lemma 2.1, we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ33_HTML.gif
(2.32)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ34_HTML.gif
(2.33)

Hence the conclusion of Lemma 2.3 is proved.

The Parseval formula for the Fourier transform together with inequality (2.18), there holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ35_HTML.gif
(2.34)

This implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq74_HTML.gif , which is Fourier transform of exact data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq75_HTML.gif , must decay rapidly at high frequencies since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq76_HTML.gif . But such a decay is not likely to occur in the Fourier transform of the measured noisy data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq77_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq78_HTML.gif . So, small perturbation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq79_HTML.gif in high frequency components can blow up and completely destroy the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq80_HTML.gif given by (2.17) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq81_HTML.gif .

3. Wavelet Dual Least Squares Method

3.1. Dual Least Squares Method

A general projection method for the operator equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq83_HTML.gif is generated by two subspace families https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq85_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq86_HTML.gif and the approximate solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq87_HTML.gif is defined to be the solution of the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ36_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq88_HTML.gif denotes the inner product in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq89_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq90_HTML.gif and subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq91_HTML.gif are chosen in such a way that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ37_HTML.gif
(3.2)
Then we have a special case of projection method known as the dual least squares method. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq92_HTML.gif is an orthogonal basis of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq94_HTML.gif is the solution of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ38_HTML.gif
(3.3)
then the approximate solution is explicitly given by the expression
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ39_HTML.gif
(3.4)

3.2. Shannon Wavelets

In [22], the Shannon scaling function is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq95_HTML.gif and its Fourier transform is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ40_HTML.gif
(3.5)
The corresponding wavelet function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq96_HTML.gif is given by its Fourier transform
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ41_HTML.gif
(3.6)
Let us list some notation: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq101_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq102_HTML.gif , the index set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ42_HTML.gif
(3.7)
Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq103_HTML.gif , hence we can define the subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq104_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ43_HTML.gif
(3.8)
Define an orthogonal projection https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq105_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ44_HTML.gif
(3.9)
then from (3.4) we easily conclude https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq106_HTML.gif . From the point of view of an application to the problem (1.1), the important property of Shannon wavelets is the compactness of their support in the frequency space. Indeed, since
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ45_HTML.gif
(3.10)
it follows that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq107_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ46_HTML.gif
(3.11)

From (3.9), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq108_HTML.gif can be seen as a low-pass filter. The frequencies with greater than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq109_HTML.gif are filtered away.

Theorem 3.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq110_HTML.gif is the solution of problem (1.1) satisfying the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq111_HTML.gif , then for any fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq112_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ47_HTML.gif
(3.12)

Proof.

From (3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ48_HTML.gif
(3.13)
Due to Parseval relation and (2.8), (2.19), and (2.1), there holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ49_HTML.gif
(3.14)

Hence the conclusion of Theorem3.1 is proved.

4. Error Estimates via Dual Least Squares Method Approximation

Before giving error estimates, we present firstly subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq113_HTML.gif . According to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq114_HTML.gif , the subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq115_HTML.gif are spanned by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq116_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ50_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq117_HTML.gif can be determined by solving the following parabolic equation (see Lemma 2.3):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ51_HTML.gif
(4.2)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq118_HTML.gif is compact, the solution exists for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq119_HTML.gif . Similarly the solution of the adjoint equation is unique. Therefore for a given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq121_HTML.gif can be uniquely determined according to (4.2), furthermore
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ52_HTML.gif
(4.3)
The approximate solution for noisy data https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq122_HTML.gif is explicitly given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ53_HTML.gif
(4.4)

Now we will devote to estimating the error https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq123_HTML.gif .

Theorem 4.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq124_HTML.gif is noisy data satisfying the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq125_HTML.gif , then for any fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq126_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ54_HTML.gif
(4.5)

Proof.

From (4.3), we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq127_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq128_HTML.gif given by (4.4), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq129_HTML.gif given by (3.4) and (2.18), for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq130_HTML.gif , there holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ55_HTML.gif
(4.6)

Hence the conclusion of Theorem 4.1 is proved.

The following is the main result of this paper.

Theorem 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq131_HTML.gif be the exact solution of (1.1) and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq132_HTML.gif be given by (4.4). If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq134_HTML.gif is such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ56_HTML.gif
(4.7)
then for any fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq135_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ57_HTML.gif
(4.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq136_HTML.gif .

Proof.

Combining Theorem 4.1 with Theorem 3.1, and noting the choice rule (4.7) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq137_HTML.gif , we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ58_HTML.gif
(4.9)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ59_HTML.gif
(4.10)
thus, there holds, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq138_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ60_HTML.gif
(4.11)

Hence the conclusion of Theorem 4.2 is proved.

Remark 4.3.
  1. (i)
    When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq140_HTML.gif , estimate (4.8) is a Hölder stability estimate given by
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ61_HTML.gif
    (4.12)
     
  1. (ii)

    When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq141_HTML.gif , estimate (4.8) is a logarithmical Hölder stability estimate.

     
  2. (iii)
    When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq142_HTML.gif , estimate (4.3) becomes
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ62_HTML.gif
    (4.13)
     

This is a logarithmical stability estimate.

Remark 4.4.

In general, the a-priori bound https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq143_HTML.gif is unknown in practice, in this case, with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ63_HTML.gif
(4.14)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ64_HTML.gif
(4.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq144_HTML.gif .

Declarations

Acknowledgments

The work is supported by the National Natural Science Foundation of China (No. 10671085), the Hight-level Personnel fund of Henan University of Technology (2007BS028), and the Fundamental Research Fund for Natural Science of Education Department of Henan Province of China (No. 2009B110007).

Authors’ Affiliations

(1)
College of Science, Henan University of Technology
(2)
School of Mathematics and Statistics, Lanzhou University

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Copyright

© W. Cheng and C.-L. Fu. 2009

This article is published under license to BioMed Central Ltd. 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.