Volume 2009, Article ID 260941,13pages doi:10.1155/2009/260941
Research Article
Solving the Axisymmetric Inverse Heat Conduction
Problem by a Wavelet Dual Least Squares Method
Wei Cheng
1and Chu-Li Fu
21College of Science, Henan University of Technology, Zhengzhou 450001, China 2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Correspondence should be addressed to Chu-Li Fu,[email protected]
Received 17 August 2008; Revised 23 January 2009; Accepted 10 March 2009
Recommended by Ugur Abdulla
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 solutionif it existsdoes 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.
Copyrightq2009 W. Cheng and C.-L. 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.
1. Introduction
Inverse heat conduction problemsIHCPhave become an interesting subject recently, and many regularization methods have been developed for the analysis of IHCP 1–13. These methods include Tikhonov method 1, 2, mollification method 3, 4, optimal filtering method 5, lines method 6, wavelet and wavelet-Galerkin method 7–11, 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 domain14–
19.
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.
measurement equipment of temperatureis installed inside the cylinderat the radiusr1,
0 < r1 < R. The correspondingly mathematical model of our problem can be described by
the following axisymmetric heat conduction problem:
∂u
∂t Δu ∂2u ∂r2
1
r ∂u
∂r, 0< r≤R, t >0, ur,0 0, 0≤r≤R,
ur1, t gt, t≥0, ur, t bounded inr0, t >0,
1.1
where the functions ur,· and g·belong to L20,∞ for every fixedr ∈ 0, R, r is the radial coordinate,gtdenotes the temperature history at one fixed radius r10 < r1 < R
of cylinder. We want to recoverur,·forr1 < r ≤ R. This problem is ill-posed problem; a
small perturbation in the data may cause dramatically large errors in the solutionur,· The details can be seen inSection 2.
To the authors’ knowledge, up to now, there is no regularization theory with error estimate for problem1.1in the intervalr1< r ≤R. The major objective of this paper is to do
the theoretic stability and convergence estimates for problem1.1.
Xiong and Fu 11 and Regi ´nska 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 problem1.1in 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 problem1.1inL2Rwith respect to variablet, we extendur ,·, g· : ur1,·, f·:uR ,·, and other functions of variabletappearing in the paper to be zero for t <0. Throughout the paper, we assume that for the exactgthe solutionuexists and satisfies an apriori bound
f·
p:uR,·p≤E, p≥0, 2.1
wheref·pis defined by
f·
p:
∞
−∞
1ξ2p fξ2dξ 1/2
. 2.2
Sinceg is measured by the thermocouple, there will be measurement errors, and we would actually have as data some functiongδ∈L2R, for which
where the constantδ >0 represents a bound on the measurement error, and · denotes the
L2Rnorm and
hξ √1
2π
∞
−∞e
−iξthtdt 2.4
is the Fourier transform of functionht. The problem1.1can be formulated, in frequency space, as follows:
iξur, ξ ∂ 2ur, ξ
∂r2
1
r
∂ur, ξ
∂r , r ∈0, R, ξ∈R, 2.5
ur1, ξ gξ, ξ∈R, 2.6
u0, ξ<∞, ξ∈R. 2.7
Then we have the following lemma.
Lemma 2.1. Problem2.5–2.7has the solution given by
ur, ξ I0
iξr I0
iξr1
gξ, r∈0, R, ξ∈R, 2.8
whereI0zdenotes modified spherical Bessel function which given by [21]
I0z ∞
k0
1
k!2
z
2
2k
. 2.9
Proof. Due to21, we can solve2.5, in the frequency domain, to obtain
ur, ξ AξI0
iξrBξK0
iξr ξ∈R, 2.10
whereK0zdenotes also modified spherical Bessel function which is given by
K0z −I0z
lnz
2 C
∞
k1
1
k!2
11
2 · · ·
1
k z
2
2k
. 2.11
Combining limz→0K0z ∞with condition2.7, we obtainBξ 0, that is,
ur, ξ AξI0
iξr, r∈0, R, ξ∈R. 2.12
According to21, there holds
I0
iξrber|ξ|riσbei|ξ|r
∞
k0
|ξ|r/24k
k!22k! 1/2
whereσsgnξ, both berxand beixdenote the Kelvin functions. Since 0≤r≤R,|ξ| ≥0, we have
∞
k0
|ξ|r/24k
k!22k! 1
|ξ|r/28
2!24!
|ξ|r/212
3!26! · · · ≥1. 2.14
Therefore, for 0≤r≤R, ξ∈R,
I0
iξr
∞
k0
|ξ|r/24k
k!22k! 1/2
≥1. 2.15
Solving the systems2.6and2.12using2.15we get
Aξ I−1 0
iξr1
gξ. 2.16
Substitution ofAξin2.16into2.12, we obtain2.8.
Applying an inverse Fourier transform to2.8, problem1.1has the solution
ur, t √1
2π
∞
−∞e
iξtI0
iξr I0
iξr1
gξdξ, r, t∈0, R×R. 2.17
In order to obtain ill-posedness of problem1.1forr, t ∈ r1, R×R, we need the
following lemma.
Lemma 2.2. If function |I0
iξr| satisfies 2.15, then there exist positive constants ck, k
1,2,3,4,such that, forr ∈r1, R
c1exp
|ξ|/2r−r1
≤I0
iξr I0
iξr1
≤c2exp
|ξ|/2r−r1
, ξ∈R, 2.18
c3exp
|ξ|/2r−R
≤I0
iξr I0
iξR
≤c4exp
|ξ|/2r−R
, ξ∈R. 2.19
Proof. First, due to21and2.15, we have, forr ∈r1, Rand|ξ| → ∞,
I0
iξrber2|ξ|rbei2|ξ|r1/2 exp
|ξ|/2r
2πr|ξ|
1O
1
then there exist positive constantsck, k1,2,3,4,such that, for|ξ|large enough, say|ξ| ≥ξ0
c1
exp|ξ|/2r
2πr|ξ|
≤I0
iξr≤c2
exp|ξ|/2r
2πr|ξ|
, r∈r1, R,
c3
exp|ξ|/2r1
2πr1
|ξ|
≤I0
iξr1≤c4
exp|ξ|/2r1
2πr1
|ξ| .
2.21
From these we know that there exist positive constantsc5andc6such that, forr ∈r1, Rand
|ξ| ≥ξ0,
c5exp
|ξ|/2r−r1
≤I0
iξr I0
iξr1
≤c6exp
|ξ|/2r−r1
. 2.22
Then, since function|I0
iξr/I0
iξr1|is continuous in the closed regionr1, R×−ξ0, ξ0.
Threrfore, there exist constantsc7andc8such that, forr∈r1, Rand|ξ| ≤ξ0,
c7exp
|ξ|/2r−r1
≤I0
iξr I0
iξr1
≤c8exp
|ξ|/2r−r1
. 2.23
Finally, combining inequalities2.22with2.23, we can see that there exist others constants
c1 and c2 such that, for r ∈ r1, R, inequalities 2.18 are valid. Similarly, we obtain
inequalities2.19.
In order to formulate problem1.1forr1 < r≤Rin terms of an operator equation in
the spaceXL2R, we define an operatorK
r :ur,· →g·, that is,
∀ur,·∈X, Krur, t gt, r1< r ≤R. 2.24
From2.8, we have
Krur, ξ
I0
iξr1
I0
iξr ur, ξ gξ. 2.25
DenoteKrur, ξ : Krur, ξ , and we can see thatKr : L2R → L2Ris a multiplication
operator:
Krur, ξ
I0
iξr1
I0
iξr ur, ξ. 2.26
Lemma 2.3. LetKr∗ be the adjoint toKr, thenK∗r corresponds to the following problem where the
left-hand side∂u/∂tof problem1.1is replaced by−∂U/∂t, says
−∂U
∂t ΔU ∂2U
∂r2
1
r ∂U
∂r , 0< r ≤R, t≥0, Ur,0 0, 0≤r ≤R,
Ur1, t gt, t≥0, Ur, t bounded inr0, t >0,
2.27
K∗r
I0
iξr1
I0
iξr
. 2.28
Proof. Via the the following relations, combining with2.26,
Kru, υ
Kru, υ
u, Kr
∗
υu, K∗rυu, K∗rυ, 2.29
we can get the adjoint operatorK∗r ofKr in frequency domain
K∗r Kr
∗
I0
iξr1
I0
iξr
. 2.30
On the other hand, the problem 2.27 can be formulated, in frequency space, as follows:
−iξUr, ξ ∂ 2Ur, ξ
∂r2
1
r
∂Ur, ξ
∂r , r∈0, R, ξ∈R,
Ur1, ξ gξ, ξ∈R,
U0, ξ<∞, ξ∈R.
2.31
Taking the conjugate operator for problem 2.5–2.7, we realize that Ur, ξ ur, ξ . Therefore, byLemma 2.1, we conclude that
Ur, ξ ur, ξ I0
iξr I0
iξr1
that is,
gξ I0
iξr1
I0
iξr
Ur, ξ K∗rUr, ξ :K∗rU. 2.33
Hence the conclusion ofLemma 2.3is proved.
The Parseval formula for the Fourier transform together with inequality2.18, there holds
ur,·2ur,·2
∞
−∞ur,·
2 dξ
∞
−∞
gξer−r1
√
|ξ|/22
I0
iξr I0
iξr1
2 dξ
≥c21
∞
−∞
gξer−r1
√
|ξ|/22dξ.
2.34
This implies thatgξ , which is Fourier transform of exact datagt, must decay rapidly at high frequencies sincer1< r. But such a decay is not likely to occur in the Fourier transform
of the measured noisy datagδtatr r1. So, small perturbation ofgtin high frequency
components can blow up and completely destroy the solutionur, tgiven by2.17forr ∈ r1, R.
3. Wavelet Dual Least Squares Method
3.1. Dual Least Squares Method
A general projection method for the operator equationKug,K:X L2R →XL2Ris
generated by two subspace families{Vj}and{Yj}ofXand the approximate solutionuj∈Vj
is defined to be the solution of the following problem:
Kuj, yg, y, ∀y∈Yj, 3.1
where·,·denotes the inner product inX. IfVj⊂RK∗and subspacesYjare chosen in such
a way that
K∗Yj Vj. 3.2
Then we have a special case of projection method known as the dual least squares method. If
{ψλ}λ∈I
jis an orthogonal basis ofVjandyλis the solution of the equation
then the approximate solution is explicitly given by the expression
uj
λ∈Ij
g, yλ 1 kλ
ψλ. 3.4
3.2. Shannon Wavelets
In22, the Shannon scaling function isφ sinπt/πtand its Fourier transform is
φξ
1, |ξ| ≤π,
0, otherwise. 3.5
The corresponding wavelet functionψis given by its Fourier transform
ψξ
e−iξ/2, π≤ |ξ| ≤2π,
0, otherwise. 3.6
Let us list some notation:φj,kt:2j/2φ2jt−k,ψj,kt:2j/2ψ2jt−k,j, k∈Z,Ψ−1,k :φ0,k
andΨl,k:ψl,kforl≥0, the index set
I {j, k}:j, k∈Z⊂Z2, IJ
{j, k}:j −1,0, . . . , J−1;k∈Z⊂Z2. 3.7
BecauseVJ VJ−1⊕WJ−1 VJ−2⊕WJ−2⊕WJ−1 · · · V0⊕W1⊕ · · · ⊕WJ−1, hence we can
define the subspacesVJ
VJspan{Ψλ}λ∈IJ. 3.8
Define an orthogonal projectionPJ :L2R →VJ:
PJϕ
λ∈IJ
ϕ,Ψλ
Ψλ, ∀ϕ∈L2R, 3.9
then from3.4we easily conclude uJ PJu. From the point of view of an application to
the problem1.1, the important property of Shannon wavelets is the compactness of their support in the frequency space. Indeed, since
ψj,kξ 2−j/2e−i2
−jkξ
ψ2−jξ, φj,kξ 2−j/2e−i2
−jkξ
φ2−jξ, 3.10
it follows that for anyk∈Z
suppψj,k
ξ:π2j ≤ |ξ| ≤π2j1, supp φj,k
From3.9,PJ can be seen as a low-pass filter. The frequencies with greater thanπ2J1are
filtered away.
Theorem 3.1. Ifur, tis the solution of problem1.1satisfying the conditionuR,·p≤E, then for any fixedr∈r1, R
ur,·−PJur,·≤c−1 3
2J1−per−R
√
1/2π2J
E. 3.12
Proof. From3.9, we have
ur,· λ
ur,·,Ψλ
Ψλ,
PJur,·
λ∈IJ
ur,·,Ψλ
Ψλ.
3.13
Due to Parseval relation and2.8,2.19, and2.1, there holds
ur,·−PJur,·ur,·−PJur,·
λ∈I
u,Ψλ Ψλ−
λ∈IJ
u,Ψλ Ψλ
λ∈Ij≥J1
u,Ψλ Ψλ
λ∈Ij≥J1 I0
i·r I0
i·r1
!
g·,Ψλ
" Ψλ
λ∈Ij≥J1 I0
i·r I0
i·R
!
f·,Ψλ
" Ψλ ≤ sup
π2J≤|ξ|≤π2J1
|ξ|−p
I0 iξr I0 iξR
λ∈Ij≥J1
#
1 ·2p/2f·,Ψλ
$ Ψλ ≤ sup
π2J≤|ξ|≤π2J1
c4ξ−per−R
√
|ξ|/2E≤c 4
2J1−per−R
√
1/2π2J
E.
3.14
Hence the conclusion of Theorem3.1is proved.
4. Error Estimates via Dual Least Squares Method Approximation
Before giving error estimates, we present firstly subspacesYj. According toK∗Yj Vj, the
subspacesYjare spanned byρλ, λ∈IJ, where
K∗ρλ Ψλ, kλρλ−1, yλ
ρλ
ρλcan be determined by solving the following parabolic equationseeLemma 2.3:
−∂U
∂t ΔU ∂2U
∂r2
1
r ∂U
∂r , 0< r ≤R, t≥0, Ur,0 0, 0≤r≤R,
Ur1, t Ψj,kt, t≥0,
Ur, t bounded inr0, t >0.
4.2
Since suppψj,kis compact, the solution exists for anyt∈0,∞. Similarly the solution of the
adjoint equation is unique. Therefore for a givenΨλ,ρλcan be uniquely determined according
to4.2, furthermore
ρλ I0 iξr I0 iξr1
Ψλξ⇐⇒yλ
I0 iξr I0 iξr1
kλΨλξ, λ{j, k}. 4.3
The approximate solution for noisy datagδis explicitly given by
PJuδr, t uδJ
λ∈IJ
uδ,Ψλ
Ψλ
λ∈IJ
gδ, yλ
1
kλΨλ. 4.4
Now we will devote to estimating the errorPJuδ−PJu.
Theorem 4.1. If gδ is noisy data satisfying the condition g·−gδ· ≤ δ, then for any fixed
r∈r1, R
PJuδ−PJu ≤c4er−r1
√
1/2π2J
δ. 4.5
Proof. From4.3, we haveyλ I0
iξr/I0
iξr1kλΨλ. Note thatPJuδgiven by4.4,PJu
given by3.4and2.18, forr1< r ≤R, there holds
PJuδr,·−PJur,·
λ∈IJ
gδ−g, yλ
1
kλΨλ
λ∈IJ
gδ−g, yλ
1 kλ Ψλ
λ∈IJ
gδ−g,
I0
i·r I0
i·r1
kλΨλ
" 1 kλ Ψλ ≤ sup
π2J−1≤|ξ|≤π2J
I0 iξr I0 iξr1 ·
λ∈IJ
gδ−g, Ψλ Ψλ
≤ sup
π2J−1≤|ξ|≤π2J
I0
iξr I0
iξr1
· PJgδ−g
≤ sup
π2J−1≤|ξ|≤π2J
I0
iξr I0
iξr1
·δ
≤c2 sup
π2J−1≤|ξ|≤π2J
er−r1
√
|ξ|/2δ
≤c2er−r1
√
1/2π2J
δ.
4.6
Hence the conclusion ofTheorem 4.1is proved.
The following is the main result of this paper.
Theorem 4.2. Letube the exact solution of 1.1and letPJuδbe given by4.4. Ifg−gδ ≤δand
JJδis such that
J log2
2
π
1
R−r1
ln
E δ
lnE
δ
−2p 2
, 4.7
then for any fixedr∈r1, R
ur,·−PJuδr,·
≤E1−R−r/R−r1δR−r/R−r1
lnE
δ
−2p1−R−r/R−r1
Co1 forδ−→0,
4.8
whereC c4R−r12pc2.
Proof. CombiningTheorem 4.1withTheorem 3.1, and noting the choice rule4.7ofJ, we can obtain
ur,·−PJuδr,·
≤c4
2J1−per−R√1/2π2J
Ec2er−r1
√
1/2π2J
δ
≤c4E
R−r1
2p
ln
E δ
lnE
δ
−2p !−2p E
δ
lnE
δ
−2p!r−R/R−r1
c2δ E δ
lnE
δ
−2p!r−r1/R−r1
≤E1−R−r/R−r1δR−r/R−r1
lnE
δ
−2p1−R−r/R−r1% c4R−r1lnE/δ2p
lnE/δlnE/δ−2p2p
c2
&
.
Note that
lnE/δ
lnE/δlnE/δ−2p
lnE/δ
lnE/δ−2plnlnE/δ −→1 forδ−→0, 4.10
thus, there holds, forδ → 0
ur,·−PJuδr,·
≤E1−R−r/R−r1δR−r/R−r1
lnE
δ
−2p1−R−r/R−r1 c4
R−r1
2p
c2o1
.
4.11
Hence the conclusion ofTheorem 4.2is proved.
Remark 4.3. iWhenp0 andr1 < r < R, estimate4.8is a H ¨older stability estimate given
by
ur,·−PJuδr,·≤c
4c2E1−R−r/R−r1δR−r/R−r1. 4.12
iiWhenp >0, r1< r < R, estimate4.8is a logarithmical H ¨older stability estimate. iiiWhenp >0, rR, estimate4.3becomes
uR,·−uδR,·≤E
lnE
δ
−2p
c4R−r12pc2o1
−→0 forδ−→0. 4.13
This is a logarithmical stability estimate.
Remark 4.4. In general, the a-priori boundEis unknown in practice, in this case, with
Jlog2
2
π
1
R−r1ln
1
δ
ln1
δ
−2p 2
, 4.14
then
ur,·−PJuδr,·≤δR−r/R−r1
ln1
δ
−2p1−R−r/R−r1
Co1 forδ−→0, 4.15
whereCc4R−r12pEc2.
Acknowledgments
Fundamental Research Fund for Natural Science of Education Department of Henan Province of ChinaNo. 2009B110007.
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