NUMERICAL MATHEMATICS
English
Series
Vol.14, No.2
A Journal of Chinese Universities
May 2005
SOLUTION OF BACKWARD HEAT PROBLEM
BY MOROZOV DISCREPANCY PRINCIPLE
AND CONDITIONAL STABILITY∗
Li Hui(
Abstract
)
)
Liu Jijun(
Consider a 1-D backward heat conduction problem with Robin boundary
condition. We recover u(x, 0) and u(x, t0 ) for t0 ∈ (0, T ) from the measured data u(x, T )
respectively. The first problem is solved by the Morozov discrepancy principle for which
a 3-order iteration procedure is applied to determine the regularizing parameter. For
the second one, we combine the conditional stability with the Tikhonov regularization
together to construct the regularizing solution for which the convergence rate is also
established. Numerical results are given to show the validity of our inversion method
Key words
Backward heat problem, regularization, conditional stability, convergence,
numerics.
AMS(2000)subject classifications 35R30, 35J05
1
Introduction
Consider the following 1-D heat conduction problem
⎧
∂u
∂
∂u
⎪
⎪
=
(k(x) ),
(x, t) ∈ (a, b) × (0, T ),
⎪
⎪
∂t
∂x
∂x
⎪
⎨
ux (a, t) − h(a)u(a, t) = 0, t ∈ (0, T ),
⎪
⎪
ux (b, t) + h(b)u(b, t) = 0, t ∈ (0, T ),
⎪
⎪
⎪
⎩ u(x, 0) = g(x),
x ∈ (a, b)
(1.1)
with positive k(x) and h(a), h(b) ≥ 0, which describes the heat conduction within a stick without
heat sink or heat source. The classical direct problem is to find u(x, t) for t > 0 from given initial
temperature g(x). This direct problem can be solved by difference schemes such as Crank-
∗
Correspondence author, e-mail: [email protected].
Received: Dec. 28, 2004.
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 181 ·
Nicolson scheme or implicit Euler scheme.
However, in many practical application areas such as archeology, it is also necessary to find
the temperature u(x, t) for t ∈ [0, T ) from the known final value u(x, T ). This is the so-called
backward heat problem which is well-known to be ill-posed, namely, the solution does not depend
continuously on the input data u(x, T ). In fact, the rapid decay of temperature with respect
to time t results in the quick disappearance of characteristic of g(x). Therefore the numerical
recovery of initial temperature from measured data at time T > 0 is very difficult due to the
input data error and computational error.
There have been much research works on the backward heat problem, for example, see
[1,2,3,4,11] and the references therein. In [1], R.Chapko used the regularized Newton method to
discuss a 2-D heat conduction problem of recovering the medium boundary, while H.Han considered the 1-D backward heat problem by the minimum energy technique and the boundary
element method([2]). In 1999, Muniz([3]) tested three numerical methods for solving 1-D backward heat problem with Dirichlet boundary condition. The first method converts the problem
to an integral equation of the first kind and solves it directly. Of course, the results are not satisfactory. In his second method, the backward implicit Euler scheme is used to get satisfactory
numerical results for exact input data. Finally he used the Tikhonov regularization method to
this problem and obtained very good numerical results. In [4,5,10], the Tikhonov regularization
is used to consider the 2-D backward heat problems with Dirichlet boundary condition. On the
choice of regularization parameter for the Tikhonov method, except for the well-known strategy, some new techniques have been developed([6],[7]). For example, the conditional stability is
applied to give a choice strategy of regularization parameter and analyze the convergence rate.
In this paper, we consider the 1-D backward heat conduction problem described by (1.1).
Our problem is to determine u(x, t0 ) for t0 ∈ [0, T ) from the noisy data of u(x, T ). We apply the
Morozov discrepancy principle to recover u(x, 0). For t0 ∈ (0, T ), the conditional stability result
is applied to construct the regularizing solution of u(x, t0 ) for which the convergence rate is also
given. It is interesting that the convergence rate for t0 ∈ (0, T ) is invalid for t0 = 0.
This paper is organized as follows. Firstly we convert this problem into an integral equation
and show the ill-posedness of the problem by logarithmic convex method in section 2. Then we
apply the Morozov discrepancy principle to recover u(x, 0) and construct an iteration procedure
with 3-order convergence rate for the regularizing parameter in section 3. In section 4, we
construct the regularizing solution of u(x, t0 ) for t0 > 0 by the conditional stability result.
Finally we give some numerical results in section 5 to show the validity of our inversion method.
· 182 ·
Li Hui
2
Liu Jijun
Ill-posedness of backward heat problem
2
∞
Denote by {un (x)}∞
n=1 ⊂ L (a, b), {λn }n=1 the eigenfunctions and eigenvalues of the prob-
lem
⎧
∂
∂v
⎪
⎪
⎨ ∂x (k(x) ∂x ) + λv(x) = 0, x ∈ (a, b),
t > 0,
⎪ vx (a) − h(a)v(a) = 0,
⎪
⎩
vx (b) + h(b)v(b) = 0,
t>0
(2.1)
respectively, where {un }∞
n=1 is the unitary orthogonal functions. Then the solution to (1.1) can
be expressed as
u(x, t) =
with the coefficients cn =
has the expression
a
∞
cn un (x) exp(−λn t)
n=1
b
g(x)un (x)dx. So the operator A : g(x) → u(x, T ) defined by (1.1)
∞
b
Ag(x) =
a n=1
un (x)un (y) exp(−λn T )g(y)dy
(2.2)
from which the equation Ag(x) = u(x, T ) can be converted as
b
K(x, y)g(y)dy = u(x, T )
(2.3)
a
with the kernel function
K(x, y) =
∞
un (x)un (y) exp(−λn T ).
n=1
It is obvious that A is a self-adjoint compact operator from L2 (a, b) to itself. On the other hand,
the coefficients cn can also be given by
cn = exp(λn T )
a
b
u(x, T )un (x)dx,
therefore
g(x) = A−1 u(x, T ) =
∞
cn un (x).
(2.4)
n=1
The determination of g(x) from this expression is unstable. That is, the small perturbation of
u(x, T ) can cause large change of g(x) given by (2.4). This fact can be seen from the following
Lemma 2.1 Define
M = h(a)k(a)u2 (a, T ) + h(b)k(b)u2 (b, T ) +
a
Then it follows that
g(·)2L2 ≥ u(·, T )2L2 exp
b
k(x)(∂x u(x, T ))2 dx.
2T M
u(·, T )2L2
.
(2.5)
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 183 ·
Proof By defining t = T − t and w(x, t ) = u(x, T − t ), the original problem is converted
into determining w(x, T ) = u(x, 0) = g(x) from
⎧
∂w
∂w
∂
⎪
⎪
),
= − (k(x)
⎪
⎪
∂t
∂x
∂x
⎪
⎨
wx (a, t ) − h(a)w(a, t ) = 0,
⎪
⎪
wx (b, t ) + h(b)w(b, t ) = 0,
⎪
⎪
⎪
⎩ w| = u(x, T ),
(x, t ) ∈ (a, b) × (0, T ),
0 < t ≤ T,
x ∈ (a, b).
t =0
Introduce H(t ) = w(·, t )2L2 =
a
b
|w(x, t )|2 dx, then it follows
2
2
H (t ) = 2 h(b)k(b)w (b, t ) + h(a)k(a)w (a, t ) +
(2.6)
0 < t ≤ T,
b
a
k(x)(wx (x, t ))2 dx
(2.7)
from (2.6), which yields
H(t )H (t ) − (H (t ))2 = 4
b
w2 dx
a
b
a
wt2 dx −
a
b
2
wwt dx
≥0
from integrating by parts and Schwartz inequality. That is, ln H(t ) is a convex function. This
fact tells us
ln H(t ) − ln H(0) ≥
ln
H(t)
,
t
t=0
t H (0)
,
H(t ) ≥ H(0) exp
H(0)
which leads to (2.5) from (2.7) and w(x, 0) = u(x, T ), w(x, T ) = g(x) by taking t = T .
Remark 2.2 We know from (2.5) that g(·) may be very large even if u(·, T ) is small,
so the determination of g(x) approximately from the noisy data of u(·, T ) should introduce
the regularization method. Obviously, this observation is also true for recovery of u(x, t0 ) for
t0 ∈ (0, T ) from u(0, T ). That is, the backward heat problem is ill-posed.
3
Discrepancy principle for recovering u(x, 0)
One of the important method of determining the regularizing parameter in Tikhonov regularization is the Morozov discrepancy principle. Under some a-priori assumption on the exact
solution, the standard regularization theory can give the convergence rate of regularizing solution with respect to the noisy level for the measured input data([8],[9]). However, the theoretical
assumptions in the standard convergence result are hard to verify in practical application. Here
we propose a numerical iteration scheme to solve the regularizing parameter from the Morozov discrepancy principle for our problem. Our iteration scheme is of 3-order convergence rate.
Once we have determined the regularizing parameter, g(x) = u(x, 0) can be computed from the
regularizing equation.
· 184 ·
Li Hui
Liu Jijun
Assume that the known noisy data of u(·, T ) meets
uδ (·, T ) − u(·, T ) ≤ δ
and introduce the Tikhonov functional
F (g, α) = Ag − uδ 2L2 (a,b) + αg2L2 (a,b) .
(3.1)
The unique minimal element gαδ of F (g, α) over L2 (a, b) satisfies the regularizing equation
αgαδ + A2 gαδ = Auδ .
(3.2)
The Morozov discrepancy principle chooses the parameter α such that the error caused by gαδ
meets
G(α) := Agαδ − uδ − δ = 0.
(3.3)
We always consider the measured temperature field large enough compared with the error level,
that is, uδ (·, T ) ≥ δ, otherwise the measured data will be contaminated completely by the
noisy. In this case, (3.3) has a unique solution α∗ . By simple computation, we know
dg δ
dg δ
dg δ
d2 gαδ α
α
G (α) = −2
, gαδ , G (α) = −2
, gαδ − 2α α 2 + gαδ ,
,
dα
dα
dα
dα2
where gαδ ,
dgαδ d2 gαδ
,
satisfy
dα dα2
(αI + A2 )gαδ = Auδ ,
(αI + A2 )
dgαδ
= −gαδ ,
dα
(αI + A2 )
d2 gαδ
dgαδ
.
=
−2
dα2
dα
The iteration procedure to solve (3.3) is as follows:
αk+1
G(αk )G (αk )
G (αk )2 − 12 G(αk )G (αk )
G(αk )
1
= αk − ·
:= F 1(αk ).
k )G (αk )
G (αk ) 1− G(α
2G (αk )2
= αk −
(3.4)
We can show that this scheme is at least 3-order convergence, that is,
|αk+1 − α∗ |
= C > 0,
k→∞ |αk − α∗ |3
lim
where α∗ is the unique exact solution to (3.3).
The following theorem on the p−order convergence is well-known:
Lemma 3.1 For the iteration scheme xk+1 = ϕ(xk ) with k = 1, 2, 3, · · · , if ϕ(x) is p−order
differentiable in the neighborhood of x∗ and
ϕ(k) (x∗ ) = 0,
k = 1, 2, 3, · · · , p − 1,
ϕ(p) (x∗ ) = 0,
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 185 ·
then the iteration scheme is p−order convergence to x∗ locally.
Based on this Lemma, we obtain
Theorem 3.2 The iteration scheme (3.4) converges at least 3−order to α∗ .
Proof For the iteration αk+1 = F 1(αk ), introduce
t(α) =
G(α)G (α)
,
G (α)2
T (α) =
1
1−
,
1
2 t(α)
s(α) =
G(α)
,
G (α)
then F 1(α) = α− s(α)T (α). Noticing G(α∗ ) = 0, the simple computations for s (α), s (α), T (α)
tell us
F 1 (α∗ ) = 1 − T (α∗ )s (α∗ ) = 0,
F 1 (α∗ ) = −s (α∗ )T (α∗ ) − 2s (α∗ )T (α∗ ) =
G (α∗ )
G (α∗ )
− 2 ∗ = 0.
∗
G (α )
2G (α )
In a similar way, we know,
F 1 (α∗ ) =
3 ∗ 2
2 G (α )
− G (α∗ )G (α∗ )
.
G (α∗ )2
Since we do not know whether F 1 (α∗ ) = 0, Lemma 3.1 means that the iteration is at least
3−order convergent.
Once determining the approximate regularizing parameter from this iteration, we can reconstruct g(x) from (3.2). The advantage of this method lies in the fact that we obtain a convergent
iteration scheme without any smooth assumption on the exact solution, which is hard to verify.
The numerical performance will be given in the last section to show the validity of this scheme.
4
Reconstruction of u(x, t0 ) for t0 ∈ (0, T )
Now we consider the recovery of u(x, t0 ) for t0 ∈ (0, T ) in this section. Contrast to the
recovery of u(x, 0), we can establish the convergence rate of regularizing solution with the aid of
conditional stability on the backward heat problem. This result makes us to give a-priori choice
strategy for the regularizing parameter and to estimate the convergence rate of approximate
solution.
We firstly establish the conditional stability result for the backward heat problem. For given
constant E > 0, introduce a set
P (E) := {g(x) ∈ L2 (a, b), g(·)L2 ≤ E}.
Lemma 4.1 For the initial value g(x) ∈ P (E), the solution to (1.1) at t0 ∈ (0, T ) meets
t0
t0
u(·, t0 )L2 ≤ E 1− T u(·, T )LT2 .
· 186 ·
Li Hui
Liu Jijun
Proof The proof is the standard logarithmic convex argument analog to that applied in
the proof of Lemma 2.1, also see [5].
A direct result of this Lemma due to the linearity of direct problem is the following conditional stability:
Lemma 4.2 For gi (x) ∈ P (E) with i = 1, 2, the solution to (1.1) meets
t0
t0
u1 (·, t0 ) − u2 (·, t0 )L2 ≤ (2E)1− T u1 (·, T ) − u2 (·, T )LT2 .
(4.1)
It is easy to see that (4.1) is also true at t0 = 0, T but trivial from which we can not get
any stability result. However, we can give an estimate on the approximation uδ (·, t0 ) obtained
from the regularizing solution of g(x) if we restrict g(x) ∈ P (E). That is,
Theorem 4.3 Assume g(x) ∈ P (E) and denote by gδδ2 the regularizing solution of (3.1)
with α = δ 2 . If we solve direct problem (1.1) with initial temperature gδδ2 to construct uδ (x, t0 )
for t0 ∈ (0, T ), then
t0
uδ (·, t0 ) − u(·, t0 )L2 ≤ 2(E + 1)(E + 2)δ T .
(4.2)
Proof For α = δ 2 , it is easy to see for the exact initial value g(x) that
F (g, δ 2 ) =
Ag(·) − uδ (·, T )2 + δ 2 g(·)2
=
u(·, T ) − uδ (·, T )2 + δ 2 g(·)2
≤
δ 2 + δ 2 g(·)2 ≤ (E 2 + 1)δ 2 ≤ (E + 1)2 δ 2 .
On the other hand, since gαδ is the minimal element of Tikhonov functional F (·, δ 2 ) in L2 (a, b),
which can be solved from (3.2) with α = δ 2 , we get
F (gαδ , δ 2 ) ≤ F (g, δ 2 ) ≤ (E + 1)2 δ 2 ,
which means
Agαδ (·) − uδ (·, T ) ≤ (E + 1)δ, gαδ (·) ≤ E + 1
(4.3)
from the definition of F (·, δ 2 ). So it follows
Agαδ (·) − Ag(·) ≤ Agαδ (·) − uδ (·, T ) + uδ (·, T ) − Ag(·) ≤ (E + 2)δ.
Since uδ (·, 0) = gδδ2 (·) ≤ E + 1, u(·, 0) = g(·) ≤ E + 1, the conditional stability result
(4.1) leads to
uδ (·, t0 ) − u(·, t0 )
t0
≤
2(E + 1)Agαδ (·) − Ag(·) T
≤
2(E + 1)(E + 2) T δ T
≤
2(E + 1)(E + 2)δ T .
t0
t0
t0
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 187 ·
The proof is complete.
Remark 4.4 In this theorem, for the regularizing solution gδδ2 of exact initial value g(x),
we do not know the convergence rate. However, with the help of conditional stability, we can
construct uδ (·, t0 ) for t0 ∈ (0, T ) from gδδ2 and obtain its Hölder type convergence rate to u(·, t0 ).
In fact, when recovering u(x, t0 ) approximately, we can see from this result that the smaller
T − t0 is, the faster the convergence rate is. This is reasonable. However, (4.2) tells us nothing
about the convergence rate for t0 = 0.
5
Numerical implementations
In this section, we give the numerical performance of recovering the initial temperature by
Morozov discrepancy principle, since the reconstruction of initial temperature, except for its own
interest, is also used in the recovery of u(x, t0 ) for t0 > 0 in our inversion scheme. The main step
used in the method is to determine the regularizing parameter by iteration scheme. The scheme
proposed in section 3 is as follows:
1. For error level δ > 0, set initial value α0 > 0 and stopping criterion > 0, kmax ;
2. Discrete and solve the equations with respect to gαδ ,
dgαδ d2 gαδ
,
for α = αk given in section
dα dα2
3;
3. Compute G(α), G (α), G (α) in terms of the expressions in section 3;
4. Obtain αk+1 from
αk+1 = αk −
G(αk )G (αk )
;
G (αk )2 − 12 G(αk )G (αk )
5. If |αk+1 − αk | ≤ or iteration number k = kmax , stop; otherwise set k = k + 1 and goto
step 2.
Without loss of generality, take (a, b) = (0, π) and divide (0, π) by points xj = jπ/m for
j = 0, 1, · · · , m with even m. We use the composite trapezoid formula to compute the integrals
in (0, π). Then (3.2) can be written as
m
m
m
2 π
π δ
δ
aj K(x, yj )
ai K(yj , zi )g (zi ) =
ai K(x, yi )u(yi , T )
αg (x) + 2
m j=0
m i=0
i=0
(5.1)
with a0 = am = 1/2 and ai = 1 for i = 1, · · · , m − 1. Taking x = xl for l = 0, 1, · · · , m generates
the matrix equation
(αI +
π2 2
π
H ) · G = H · U,
2
m
m
(5.2)
· 188 ·
Li Hui
Liu Jijun
−3
5.5
x 10
5
α=0.9515E−03
4.5
rel err
4
3.5
3
2.5
2
1.5
1
0
1
2
α
3
4
5
−3
x 10
1: Relative error with respect to α for δ = 0.001
T
T
where G = g δ (x0 ), · · · , g δ (xm ) , U = u(y0 , T ), · · · , u(ym , T ) and
⎡
a0 K(x0 , y0 )
⎢ a0 K(x1 , y0 )
⎢
⎢
..
H =⎢
.
⎢
⎣
a0 K(xm , y0 )
Example 1
a1 K(x0 , y1 )
a1 K(x1 , y1 )
..
.
···
···
..
.
a1 K(xm , y1 ) · · ·
⎤
am K(x0 , ym )
am K(x1 , ym ) ⎥
⎥
⎥
..
⎥.
.
⎥
⎦
am K(xm , ym )
Consider the following model problem
⎧
∂2u
∂u
⎪
⎪
(x, t) ∈ (0, π) × (0, T ),
⎨ ∂t = ∂x2 ,
ux (0, t) = ux (π, t) = 0, t > 0,
⎪
⎪
⎩
u(x, 0) = cos x,
x ∈ [0, π].
(5.3)
It is easy to see that u(x, t) = e−t cos x is the exact solution of (5.3).
For this problem, the kernel function of operator A has the expression
K(x, y) =
∞
1
1+2
cos nx cos ny exp(−n2 T ) .
π
n=1
We approximate this infinite series by its first 30-terms. For the above schemes, we set α0 =
0.01, = 10−8 , kmax = 50, m = 100, T = 0.005 and take δ = 0.001, 0.01 respectively to generate
the noisy data by
uδ (x, T ) = u(x, T ) + δ sin(2x − 1).
(5.4)
For δ = 0.001, the iteration stops after 5 times with final value α∗ = 9.5151453 × 10−4 ; while
for δ = 0.01, we get final value α∗ = 8.5541046 × 10−3 after 2 iteration. The relative error
distributions with respect to different α are given in Fig.1 and Fig.2 from which we can see that
our iteration schemes indeed generate an approximate optimal α.
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 189 ·
0.018
0.017
α=8.554E−03
rel err
0.016
0.015
0.014
0.013
0.012
0.011
0
0.005
α
0.01
0.015
2: Relative error with respect to α for δ = 0.01
In this example, the final time T = 0.005 used to recover u(x, 0) is of course too small. The
next example shows we can still recover u(x, 0) for relatively large T if the diffusion process is
not too strong. Such kind of problems often arise in the area of archeology.
Example 2
Backward problem with small diffusion coefficient. Consider
⎧
2
∂u
⎪
2∂ u
⎪
(x, t) ∈ (0, π) × (0, T ),
⎨ ∂t = d ∂x2 ,
ux (0, t) = ux (π, t) + u(π, t) = 0, 0 < t ≤ T,
⎪
⎪
⎩
u(x, 0) = cos(5x),
x ∈ (0, π)
(5.5)
with d = 0.1. For this mixed boundary value problem, we apply the finite difference for the
direct problem to simulate u(x, T ) for our inverse problem. More precisely, we apply the 4-order
compact scheme at interior points and 2-order forward difference at boundary points. In this
π
case, the convergence order is O(τ 2 + h2 ). In our computation, we take spatial step ∆x =
200
and time step ∆t = 10−3 .
The kernel function K(x, y) for operator A in this case can be expressed as
K(x, y) =
with the constant
∞
1
cos λn x cos λn y exp(−λn d2 T )
L
n=1 n
(5.6)
1
π
+ √ sin 2 λn π,
2
4 λn
1
where γn is the zero points of equation ctgx − x = 0. For this kernel function given in (5.6),
π
we approximate it by the first 20-terms in the series. Then we also generate the analogy to (5.2)
γn
,
λn =
π
Ln =
from (5.1) if we discrete x.
To show the stability performance, we give the numerical results for noisy input data given
at T = 0.5 with different error level δ in Tab.5.1. The noisy data is also generated in the form
of (5.4).
· 190 ·
Li Hui
Liu Jijun
2.5
Exact
T=0.5
T=1
T=5
2
initial temperature
1.5
1
0.5
0
−0.5
−1
−1.5
0
0.5
1
1.5
2
2.5
3
3.5
x
3: Recovery of u(x, 0) from exact u(x, T ) with different T
0.2
0.15
0.1
0.05
Error
0
−0.05
−0.1
α=1E−4
α=1E−1
α=1E−14
−0.15
−0.2
−0.25
0
0.5
1
1.5
2
2.5
3
3.5
x
4: Point-wise error dependance for different α with δ = 0.005, T = 0.5
To compare our iteration results, we firstly present the recovery results with m = 200 from
exact input data for different final time T > 0, see Fig.3.
On the other hand, we also need to consider the point-wise error distribution for different
α from which we can evaluate our strategy of choosing α by iteration scheme. This dependance
is shown in Fig.4 for δ = 0.005, T = 0.5.
Tab.5.1 Recovery results with T = 0.5 for different error level δ
x=0.031
x=1.413
x=1.57
x=1.915
x=2.512
Exact Value
0.987688
0.707107
1.339745D-7
-0.987688
1.0
δ = 0.0
1.042477
0.708769
-1.817299D-3
-0.987015
0.998254
δ = 0.001
1.041568
0.709752
-9.545695D-4
-0.986705
0.997470
δ = 0.005
1.037934
0.713682
-2.496347D-3
-0.985465
0.994333
δ = 0.01
1.033391
0.718594
6.809993D-3
-0.983916
0.990411
For our iteration scheme to determine the regularizing parameter α with δ = 0.005 , we
finally get α∗ = 3.7542086 × 10−3 by 4 iterations. Here the iteration parameters are α0 =
0.01, = 10−8 , kmax = 50, m = 200. Compared this value with Fig.4, we know this value of α∗
Solution of Backward Heat Problem by Morozov Principle and Conditional Stability
· 191 ·
0.35
0.3
temperature
0.25
0.2
0.15
Exact
T=0.5
T=1
T=5
0.1
0.05
0
−0.05
0
0.5
1
1.5
2
2.5
3
3.5
x
5: Recovery result from u(x, T ) with different T
0.015
α=1E−13
α=1E−4
α=1E−2
0.01
Error
0.005
0
−0.005
−0.01
0
0.5
1
1.5
2
2.5
3
3.5
x
6: Point-wise error dependance for different α with δ = 0.005, T = 0.5
can reconstruct a satisfactory approximation of u(x, 0).
Example 3
Consider the model problem given in Example 2 with non-smooth initial value
⎧
π
⎨ 0.2x
x ∈ [0, ],
2
g(x) = u(x, 0) =
(5.7)
⎩ 0.2(π − x) x ∈ ( π , π].
2
Using the argument similar to the above two examples, we also test the inversion results for
different final value time T with exact input data and show the inversion error dependence on
different α. The results are shown in Fig.5 and Fig.6. For δ = 0.005, we get the regularizing
parameter α∗ = 1.914127 × 10−2 by 4 iterations. Obviously, the inversion results for the whole
picture as well as the choice of α are also satisfactory. Please notice, we can not identify the
non-differential point x0 = π/2 clearly due to the rapidly decay of temperature for the direct
problem.
The above three examples show the validity of our inversion method. Since we have recovered
the initial temperature g(x) successfully, the inversion scheme of recovering u(x, t0 ) for t0 ∈ (0, T )
from noisy data of u(x, T ) proposed in section 4 can be implemented efficiently, noticing the wellposedness of direct heat conduction problem. We omit the numerical realizations.
· 192 ·
Li Hui
Liu Jijun
Acknowledgement This work is supported by NSFC(No.10371018).
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Li Hui Department of Mathematics, Southeast University, Nanjing 210096, PRC.
Liu Jijun Department of Mathematics, Southeast University, Nanjing 210096, PRC.