Semi-analytic conjugate gradient method applied to a simple inverse heat conduction problem

(1)

Semi-analytic Conjugate Gradient Method applied to a

simple Inverse Heat Conduction Problem

J.-C. Jolly ^; L. Autrique ^;

jean-claude.jolly@univ-angers.fr laurent.autriquer@univ-angers.fr

University of Angers, LARIS, 62, av. Notre-Dame-du-Lac, 49000 Angers, France

Abstract: On the most simple 1-D heat equation, a simple identi…cation problem of heat

‡ux on one side from temperature measurement on the other side is solved with a conjugate gradient method (CGM). What is new in this well known and academic problem is that the CGM can be developped explicitely. This means that the CGM is given with explicit formulae into an approximation space split between a polynomial part and an exponential part. A …ltering property of the CGM is deduced. These results correct and deepen the work of previous authors.

Keywords:Conjugate gradient method, Inverse heat conduction problem, Filtering 1. INTRODUCTION

Lettf >0; !2R ;and q(t) = exp (j!t) wherej² = 1.

For(x; t)2(0;1) (0; t_f), let us consider the normalised 1-D heat equation with Newmann boundary conditions :

T^t(x; t) T^xx(x; t) = 0; T (x;0) = 0; Tx(0; t) = 0 T^x(1; t) =u(t)

(1) where u(t) =q(t)for the moment. The complex solution of this well posed PDE is [Zau:06]

T (x; t) =q(t) (x) +S(x; t) where

8>

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:

(x) = cosh p

j!x with = 1

pj!sinh p j!

S(x; t) = 2 X1 n=1

( 1)ⁿexp n² ²t² cos (n x) n² ²+j!

with = j

!(2) Because cosh is even and sinh is odd, this formulation is independent of an arbitrary but …xed choice between the two determinations of the complex square root p

j!.

The classical way to …nd (2) is to seek …rstly (x) such that j! xx = 0; x(0) = 0; x(1) = 1and secondly S(x; t)by variable separation and Fourier series such that St Sxx= 0; Sx(0; t) =Sx(1; t) = 0; (x) +S(x;0) = 0.

Now, let us setT(t) =T (0; t). This projection de…nes the operatorAtf by

T =A_t_fu: (3)

Since the initial condition in (1) is homogeneous, operator A_t_f is linear. A classical Inverse Heat Conduction Problem (IHCP) consists to determine u for givenT , i.e to look forA_t_f¹:

u =A_t_f¹T : (4)

The existence and unicity ofu can be established in suitable spaces for (1). However, since operatorA_t_f is in…nite dimensional and compact, it is an ill posed problem. It has been shown [Alif:95] that with the CGM, which is a descent method applied to the criterion J²(u) = ¹₂kT(u) T k² and which gives a minimizing sequence u^k _k

2N, it can be regularized in a Tichono¤ sense with the number k of iterations as a regularization parameter. Then, under suitable conditionsC on the perturbation of A_t_f and the choice of u⁰, there exists a stopping rule K =K(C) for the CGM that givesu^K theorically as closed as wanted to the unique solution u . Despite there is no e¢ cient way to ensure that, for a given precision, such conditions C are satis…ed, many applications much more complex than (4) have shown that the CGM can bring a signi…cant improvement. More precisely, introducing the initial ‡ux u⁰, it is frequently observed in applications that for a certain K the CGM gives a quite satisfying solution to (4), that is to say

u CGMA_tf u⁰; T ; K = ^K_A_tf_;T u⁰ =u^K (5) where A_tf;T is the discrete dynamical system underlied to the CGM and ^K_A

tf;T = A_tf;T A_tf;T . In this in…nite-dimensional framework, despite in some cases explicit solutions to (3) are well known [Zau:06], the calculus of (5) is always made via a discretization method needed for solving (3). Consequently, no explicit formulation of

A_tf;T is known. In the second section, under a natural approximation, we will present an algebraic formulation of

A_tf;T u^k . More exactly,u^k in _A_tf_;T will be replaced by u^k; p^k ¹ or u^k; p^k ¹; T^k ¹ , to be de…ned. This will be established for

u (t) = Xm i=1

iqi(t) where qi(t) = exp (j!kt);

!_i2R and _i2C;

(2)

and in the restricted case where u⁰(t) = 0. This will give a semi-analytic solution u^K expanded in the space C^2K+1[tf t] QmwhereQm=Vect_C(qi; i= 1; : : : ; m).

For lack of space, proofs will be omitted or only sketched, but precised formulae will be given which can be tested on a computer algebra system. Note that if it is known a priori that T (t)is ²_!-periodic, for example T(t)is in L² 0;²_! eventually extended to(0; tf)by periodicity, it is relevant to look foru with a spectral method, without invoking the CGM.

Last but not least, problem (5) takes its source in the previous work [Prud:98]. These authors tried to give what we call the initialization step of the CGM and to draw what will look like the next steps, without proving the solvability of the general problem. Their motivation was to establish a …ltering property of the CGM. If(!_k)_k=1;:::;m is an increasing sequence of R+, they suggested that the

…rst iterations of the CGM give a good rendering of ₁q₁, further iterations are needed to identify 2q2 and so on for the following terms kqk. Based on the foregoing semi-analytic solutionu^K and on simulations, this will be disccused in the third section.

Notations. In this paper, if M is a complex matrix then we note M^T for its transpose and M for its conjugate transpose –and therefore z is the conjugate of z in C– while if A is an operator we note A its adjoint.

Notations h; i and k k are used for the usual hermitian product and associated norm in L²(0; t_f). For a complex sequence (s0; s1; : : : ; sn) (resp. (s1; : : : ; sn)) the associated column vector of Mn+1;1(C) (resp. Mn;1(C)) is noted s^[n] (resp. s^[n]⁰). For a double complex sequence ( 1; : : : ; m);(s0; s1; : : : ; sn), the associated column vector of M^m+n+1;1(C)is noted S^[m;n] =

[m]0

s^[n] . Matrix notations ^h^mⁱ⁰; ^(m)⁰ designate diag( 1; : : : ; m)2 M^m(C) and ( 1 m)2 M1;m(C) respectively. The null matrix of M^n;p(C) is O^n;p. The notation i j = 1 if i j and i j= 0otherwise is used. Forn k 0the number of k-permutations ofnisA^k_n= _{(n k)!}^n! and the number of k-combinations of an n-set isC_n^k =^A_k!^kⁿ. For t2[0; t_f], we introduce the backward time =tf t.

2. CONJUGATE GRADIENT METHOD Following [Alif:95], two kinds of PDEs are to be solved for the CGM :

–The direct one, as it is the case for (1) : Tt^k(x; t) Txx^k (x; t) = 0;

T^k(x;0) = 0; Tx^k(0; t) = 0 Tx^k(1; t) =u(t)

(6) but in the more general case whereu(t) =Pm

i=1 k iq_i(t) + Pn

l=0r^k_l ^l, which is denoted u^k;[m;n](t), with ^k_i; r^k_l 2 C. This PDE will give not only the direct temperature T(t) , T^k;[m;n+1](t) = Au^k;[m;n](t) but also what is called in the literature the sensitivity temperatureT~(t), T~^k;[m;n+1](t) = Ap^k;[m;n](t)for the modi…ed gradient or descent direction p(t) , p^k;[m;n](t). Here, superscripts [; n];[; n+ 1]anticipate polynomial forms in of degrees

n; n+1respectively while superscriptkanticipates thek^th iteration of the CGM.

– The adjoint one which in backward time takes the form :

T^^k(x; ) T^xx^k (x; ) = 0;

Tb^k(x;0) = 0; Tbx^k(0; t) =u(t) Tbx^k(1; t) = 0

(7) forPum(t) = T^k;[m;n](t) = T^k;[m;n](t) T (t) ,

i=1 k

iq_i(t) +Pn

l=0s^k_l ^l . In direct timet, T^^k(x; t_f t) =T^k(x; t)

is the adjoint temperature at thek^thiteration of the CGM.

Anticipating the degree n in , this de…nes the adjoint operatorAby :

T^k;[m;n+1](t) =A T^k;[m;n](t)

Next, we are going to see how temperatures T^k;[m;n+1], T~^k;[n+1] and T^k;[m;n+1] can be calculated. A L² norm calculus will follow. For sake of clarity, iteration numberk will be omitted.

2.1 Direct and sensitive PDE

It can be veri…ed that the solutionT (x; t)to (6) has the form :

T (x; t) = Xm i=1

iR1i(x; t) +R2(x; t); R1i(x; t) =qi(t) i(x) +Si(x; t);

R2(x; t) =h(x; t) +H(x; t);

hpolynomial of degree2 (n+ 1) forx whereR_1i; R₂; _i; S_i; h; H are solutions of

j! _i(x) _i;xx(x) = 0

Si;t Si;xx= 0 ; h_t h_xx= 0 Ht Hxx= 0 with boundary and initial conditions

i;x(0) = 0

i;x(1) = 1 ; S_i;x(0; t) = 0 Si;x(1; t) = 0 ; 8>

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>:

hx(0; t) = 0 h_x(1; t) =

Xn l=0

r_l ^l ; Hx(0; t) = 0 H_x(1; t) = 0 ;

i(x) +Si(x;0) = 0 h(x;0) +H(x;0) = 0 : This gives :

T(t) = Xm i=1

i(qi(t) i(0) +Si(0; t)) +h(0; t) +H(0; t): (8) Functions i and Si have already been given in (2).

Polynomialhis sought in the form : h(x; t) =

n+1X

i=0

Bi( )x²ⁱ It can be shown thatB_i must satisfy

Bi( ) = ( 1)^{n i+1}

n i+1X

l=0

1

l!A^2l_2(i+l)bi+l l; i= 0; : : : ; n+ 1

(3)

where b1; : : : ; bn+1 are arbitrary complex constants and b₀ can be choosen equal to 0. Then, condition h_x(1; t) = Pn

l=0rl lgives the relationDr=Cb where D= 1

2diag ((i 1)!)i=1;:::;n+1

C= ( 1)ⁿ ( 1)^i+j(j i+ 1)A²⁽ⁱ_2j ¹⁾ _j _i

i;j=1;:::;n+1

r= 0 B@

r₀ ... n

1 CA; b=

0 B@

b₁ ... bn+1

1 CA:

The key point of this resolution is that the square matrix C admits a calculable inverse which takes the form :

C ¹= ( 1)ⁿ dj i

A²⁽ⁱ_2i ¹⁾ ^j ⁱ

!

i=1;:::;n+1 j=1;:::;n+1

where the complex sequence(d_l)_l_2Nis given by the recurrence relation

d₀= 1; andd_l=

l 1

X

i=0

( 1)^l+i+1

(2 (l i) + 1)!d_i forl 1:

Thus,h(x; t)is de…ned. In particular, whe have : h(0; t) =

Xn l=1

l

0

@rl 1

l +

Xn j=l

dj l+1A^j_j ^lrj

1

A ⁿ⁺¹rn

n+ 1 (9) Theoretically,H(x; t)can then be found by separation of variables and Fourier series fromh(x;0) +H(x;0) = 0in the same way as forS(x;0)from (x) +S(x;0) = 0. This means thatH(x; t)will take the form

H(x; t) =c₀[ h(x;0)]

+ X1 n=1

exp n² ²t² an[ h(x;0)] cos (n x): However, since h(x;0) has a much more complicated form than (x)for Fourier series, coe¢ cientsan[ h(x;0)]

are practically not available. We make the important hypothesis :

Assumption 1. Let us suppose t 0 and that the following approximations are available :

Si(x; t)

t 0 i; H(x; t)

t 0c0[ h(x;0)]:

From a system point of view, this amounts to neglecting the transient part in front of the steady state part of the PDE response tou(t). Even forc0[ h(x;0)], the calculus is not trivial. We found

c0[ h(x;0)] = Xn j=0

sjrj

withs_j = 1

j+ 1t^j+1_f e_j; e_j= ( 1)^jj!

Xj i=0

( 1)ⁱ 1

(2 (j i) + 3)!d_i:

(10)

Thus a second complex sequence(e_j)_j_2Nappears. Solving (6) is achieved and, according to (2), (8), (9), (10), this leads to :

T(t) '

t 0

Xm i=1

i( _iq(t) + _i) +h(0; t) +c₀[ h(x;0)]: (11) More precisely, with T^[m;n+1](t) = Pm

i=1 iqi(t) + Pn+1

l=0 al l andu^[m;n](t) =Pm

i=1 iqi(t) +Pn

l=0rl l, this relation is equivalent to

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:

i= i i; i= 1; : : : ; m a₀=

Xm i=1

i i+c₀[ h(x;0)]

= Xm i=1

i i+ Xn

l=0

1

l+ 1t^l+1_f el rl

a_k= 1 kr_k ₁

Xn l=k

d_{l k+1}A^{l k}_l r_l for1 k n a_n+1= 1

n+ 1r_n:

Using notations of the introduction as for example T^[m;n+1]=

[m]0

a^[n+1] ; Q^[m;n] =

[m]0

r^[n] ; (12) this in turn can be put under the matrix form

[m]0

a^[n+1]

| {z }

T^[m;n+1]

= 0

@

[m]0

a₀ a^[n+1]⁰

1 A=

0

@

[m]0 O^m;n+1

(m)0 F⁽ⁿ⁾ O^n+1;m E^[n]

1 A

| {z }

A^[n]

[m]0

r^[n]

| {z }

Q^[m;n]

that is to say T^[n+1]=A^[m;n]Q^[m;n]

(13) whereF⁽ⁿ⁾= tf e0

1

2t²_f e1 0 1

n+ 1tⁿ⁺¹_f en 2 M1;n+1(Q[t_f]); E^[n] = ( _i;j)i;j=1;:::;n+12 Mn+1(Q[t_f])

8>

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>:

j;j = 1

j for1 j n+ 1

i;j= d_j _iA^j_j ⁱ₁ ¹for2 i+ 1 j n+ 1

i;j= 0forj+ 1< i; 2 i n+ 1:

2.2 Adjoint PDE

A similar solving for (7) can be achieved. We omit it and give the result below.

For u(t) , T^[m;n](t) = Pm

i=1 iqi(t) Pn i=0si i

and under an assumption similar to the above 1 but for t tf, we …nd thatT(t),T^[m;n+1](t) =Pm

i=1 iqi(t) + Pn+1

i=0 a_i ⁱ is given by relations : 8>

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>:

i= i ; i= 1; : : : ; m a0=

Xm i=1

iexp (j!itf) i+ Xn j=0

sjfj

a_l= 1 ls_l ₁+

Xn j=l

s_jC_j^j ^lf_j _lforl= 1; : : : ; n an+1= 1

n+ 1sn

(14)

which make appear another two sequences

(4)

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>>

:

f_j = 1

A^j+1_2j+1 2e_j+1+ ( 1)^jd_j+1 j!

Xj k=1

ek+ ( 1)^k ¹dk

A^2k_2j+1¹

A^j+1_2j+1 forj= 0; : : : ; n ek =

Xk i=0

( 1)ⁱ 1

(2 (k i))!di fork= 0; : : : ; n+ 1:

Relation (14) admits in turn the matrix form : a^[n+1]

| {z }

T^[n+1]

= 0

@ a₀ a^[n+1]⁰

1 A

= 0

@

hmi0 O^m;n+1 L^(m)⁰ f⁽ⁿ⁾ O^n+1;m E^[n]

1 A

| {z }

A^[n]

[m]0

s^[n]

| {z }

T^[n]

that is to say T^[n+1]=A^[n] T^[n]

(15)

where f⁽ⁿ⁾ = (f₀ f₁ 0 f_n) 2 M^1;n+1(Q); E^[n] =

i;j i;j=1;:::;n+12 Mn+1(Q) 8>

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>>

:

j;j= 1

j for1 j n+ 1

i;j=C_j^j ₁ⁱ ¹f_j _i ₁ for2 i+ 1 j n+ 1

i;j= 0for1 j < i n+ 1:

2.3 ParticularL² norms Forhu; vi=Rtf

0 u(t)v(t)dtone can easily compute : D(t_f t)^k;(t_f t)^lE

= t^k+l+1_f k+l+ 1 D

(tf t)^l; qi(t)E

= i

0 BB BB

@"li+ l 1 l 1

X

k=0

( i)^kA^k_lt^{l k}_f

| {z }

Pli

1 CC CC A

with8

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>:

"li= ( )^ll! (1 exp (j!itf)) = ( i)^ll!"0i

P_li="_li+ _l ₁

l 1

X

k=0

( _i)^kA^k_lt^{l k}_f ; P_0i="_0i which satis…esP_li=t^l_f _ilP_li ₁ ifl 1:

The L² norme of T^[m;n] represented by T^[m;n] is then given by T^{[m;n] 2} =T^[m;n] ^[m;n]T^[m;n] where ^[m;n] 2 Mm+n+1(Q[t_f])is de…ned by :

[m;n]= M^[m]⁰ ^(m)⁰ P^[n]

P^{[n] (m)}⁰ Q^[n] (16)

with Q^[n]= t^i+j_f ¹ i+j 1

!

i;j=1;:::;n+1

; P^{[n] (m)}⁰= ( iPli)l=0;:::;n

i=1;:::;m 2 M^n+1;m(Q[tf]) andM^[m]⁰ = (mi;j)i;j=1;:::;m2 M^m(C); m_i;j=

8<

:

tf ifi=j 1

!j !i

(1 exp (jt_f(!_j !_i))) ifi6=j:

2.4 Algorithm

In this academic study, the measured temperatureT (t) = Pm

i=1 i( _iq_i(t) + _i) T^[0]atx= 0has been simulated by the response of PDE (6) to u (t) = Pm

i=1 iq_i(t) under approximation 1. The criterion to be minimized is

J_k² J² u^k = 1

2 T u^k ².

Following [Alif:95] and using vectors T^k;[m;n], T^k;[m;n], Te^k;[m;n],P^k;[m;n],U^k;[m;n] instead of function T^k,T^k, Te^k, p^k,u^k according to notations of the introduction, and so on for matrices A^[m;n];A^[m;n]; ^[m;n] instead of A; A;k k² according to notations (13), (15), (16), we can explicit below the CGM where the degreen n(k)in backward time is speci…ed according tok:

(1) Initial step

Choose U⁰, that is to say …xe U⁰ = 0 in our case ; consequently, T⁰=AU⁰ is simplyT⁰= 0;

Calculate T^0;[m;0] =T⁰ T^[m;0] = T^[0] andJ₀²=

1 2 1

2T^[m;0] ^[m;0]T^[m;0] ;

CalculateT^0;[m;1]=A^[m;0] T^0;[m;0] =A^[m;0]T^[m;0]m

and set P^0;[m;1]=T^0;[m;1] ;

Calculate eT^0;[m;2] = A^[m;1]P^0;[m;1] = A^[m;1]T^0;[m;1]

and ₀=^T^0;[m;1] ^[m;1]^T^0;[m;1]

e

T^0;[m;2] ^[m;2]eT^0;[m;2] ;

Calculate U^1;[m;1]=U⁰ 0P^0;[m;1]= 0T^0;[m;1] ; (2) Heredity step

Let suppose that fork 1,U^k;[m;2k ^1]; P^k ^1;[m;2k ^1]; T^k ^1;[m;2k ^1]

are given ;

Calculate T^k;[m;2k]=A^[m;2k ^1]U^k;[m;2k ^1] ;

Calculate T^k;[m;2k] = T^k;[m;2k] T^[m;0] and J_k² =

1

2 T^k;[m;2k] ^[m;2k] T^k;[m;2k] ;

Calculate T^k;[m;2k+1]=A^[m;2k] T^k;[m;2k] ; Calculate ^k given by

k = T^k;[m;2k+1] ^k;[m;2k+1]T^k;[m;2k+1]

T^k ^1;[m;2k ^1] ^k ^1;[m;2k ^1]T^k ^1;[m;2k ^1]

and deduce

P^k;[m;2k+1]=T^k;[m;2k+1]+ ^k P^k ^1;[m;2k ^1]

O^2;1 ;

Calculate Te^k;[m;2k+2]=A^[m;2k+1]P^k;[m;2k+1]; Calculate ^k given by

k =T^k;[m;2k+1] ^k;[m;2k+1]T^k;[m;2k+1]

Te^k;[m;2k+2] ^k;[m;2k+2]Te^k;[m;2k+2]

and deduce

Uk+1;[m;2k+1]= U^k;[m;2k ^1]

O^2;1

kP^k;[m;2k+1]

(3) Recurrence principle step : whilek < K, replacekby k+ 1and return to step 2

The above algorithm de…nes the CGM discrete dynamical system (5) foru⁰= 0by :

(5)

A_{tf ;T} : (M^m+2k(C))³!(M^m+2k+2(C))³ U^k;[m;2k ^1];P^k ^1;[m;2k ^1]; T^k ^1;[m;2k ^1]

7! U^k+1;[2k+1];P^k;[2k+1]; Tk+1;[m;2k+1]

(17)

3. FILTERING PROPERTY OF THE CGM 3.1 A mitigation propertie of the CGM

The descent memory coe¢ cient ^k= ^T

k 2

T^k ¹ ² admits the alternative determination ^k = ^Ap

k 1;AT^k

kAp^k ¹k² [Alif]. Con- sequently, in the foregoing algorithm ^k_A

tf ;T u^k; p^k ¹; T^k ¹ can be substituted by ^k_A

tf ;T u^k; p^k ¹ . Let us consider u =Pm

i=1 iq_i, with q_i(t) = exp (j!_it) and i 2 C , the ‡ux to be identi…ed on the right side of the rode. Applying our CGM to such u leads us to look at sequences u^k; p^k ¹ in the form u^k = Q^k +D^k, p^k ¹ = P^k ¹ +C^k ¹ with Q^k = Pm

i=1 k

iqi, P^k ¹ = Pm

i=1 k 1

i qi and D^k; C^k ¹ polynomials terms in C[ ].

Two observations can be done. The …rst one is thatj ⁱj= j (!_i)jas de…ned in (2) is an even function of!_i, quickly decreasing towards 0 on R+. The second one, based on simulations (see Fig. 1, 2, 3), can be stated as follows : Assumption 2. TermsD^k in u^k andC^k ¹ in p^k ¹ can be roughly neglected in front ofQ^k andP^k ¹ respectively.

The following result can then be prooved :

Lemma 1. For 2 Rset f (x) = 1 + x. We have thus, for example, f ² f ¹ (1) = 1 + ² 1 + ¹ . Let ^k be de…ned by

k = Xk i=0

i+ ^{1 1}+ Xk

i=2

i i f ⁱ ¹ f ⁱ ² f ¹ (1) (18) which is allways positive. Under assumption 2, we have

k+1

i = _i j ij² ^k+o j ij² : (19) Note that ^k is independant from i. If k=K is the rank at which the CGM is stopped, the product ij ⁱj² ^K is to be compared with the expected value _i. At most one index i, or two indexesi and i⁰ if !i = !i⁰, can give a value of j ij² ^K closed to one. The …ltering property of the CGM does not occur in the sens expected by authors [Prud:98] but rather as a mitigation property.

3.2 Simulations

Three con…gurations were tested with the CGM algorithm (17) under Maple 18. For same tf = 10 and q1 = exp ( t); q₂= exp (2 t); q₃= exp (5 t), they are :

(i) u (t) = Im (q1) = sin ( t) ;

(ii) u (t) = Im (5q1+ 2q2+q3) = 5 sin ( t)+2 sin (2 t)+

sin (5 t);

(iii) u (t) = Im (q1+ 2q2+ 5q3) = sin ( t) + 2 sin (2 t) + 5 sin (5 t):

Results are shown on …gures 1, 2 and 3 obtained in about 200s for each one. The upper-left sub-…gure gives the measured temperatureT (t)(dash line) and the calculated temperatureT^K(t)(solid ligne) at x= 0, the upper-right sub-…gure gives the true ‡ux u (t) (dash line) and the calculated ‡ux u^K(t) at x = 1, the lower-left sub…gure gives the iterated residual criterion J_k², the lower-right sub-…gure gives the true ‡ux u (t) (dash line) and the polynomial partD^K(t)of the calculated ‡ux u^K(t). The regularization parameterKhas been choosen to minimize bothJ_k²=¹₂ Au^k T ² and u^k= u^k u ². In this three cases the polynomial partsD^K compared to the exponential partQ^Khas a low contribution to the total heat ‡uxu^K. This is measured by ratesk^D^Kk²

ku^Kk² more or less close to zero by upper value, see the …rst line of table (20).

Lemma 1 is illustrated by j ⁱj² ^K

i2f1;2;3g given in the second line of table (20). Its validity, linked to the rightness of the approximation of ^j ⁱ^j

j ^Kⁱ j by j ij² ^K

i2f1;2;3gcan be appreciated by comparison of the second and third lines of table (20). This third line tests the ability of the CGM to recover a sequence of increasing frequencies.

case (i) case (ii) case (iii) (!i) ( ;2 ;5 ) ( ;2 ;5 ) (c i) 1 (5;2;1) (1;2;5)

k^D^Kk²

ku^Kk² 0:059 0:053 0:12 j ⁱj² ^K 0:73

0:82 0:16 0:0085

! 0:99

0:19 0:010

!

j ⁱj

j ⁱ^Kj 0:97

0:98 0:25 0:014

! 1:09

0:29 0:017

!

(20) With a ratio ^j ¹^j

j ^K¹j between 0.97 and 1.09 (fourth line of table (20)), we can see that the CGM gives a good rendering of the lower frequency !1 = . This is not the case for upper frequencies !_i; i = 1;2 for which a ratio

j ⁱj

j ^Kⁱ j less than 0.3 shows that the identi…cation by the CGM is not null but quite poor. In cases (ii) and (iii), it is interesting to calculate the mitigation ratios ^j ⁱ⁺¹^j

j ^Kⁱ⁺¹j=^j ⁱ^j j ^Kⁱ j : mitigation ratios case (ii) case (iii)

j 2j

j ²^Kj=^j ¹^j j ^K¹j; ^j ³^j

j ³^Kj=^j ²^j

j ^K²j 0:25; 0:057 0:27; 0:058 This mitigation ratio is a decreasing function of the frequency which is, in this application, quite independant of the magnitudes ( i)_i_2f_1;2;3_g. As can be seen on the lower left sub…gures of Figs. 1, 2, 3, the divergence of the residual criterionJ_k² for higher iteration than K does not allow to hope a better recovery of the higher frequencies by the CGM.

4. CONCLUSION

Two new results have been presented here for the CGM applied to a simple IHCP. The …rst one is explicit formulae

(6)

Fig. 1.Im (u (t)) = sin ( t)

Fig. 2.Im (u (t)) = 5 sin ( t) + 2 sin (2 t) + sin (5 t) given in (13), (15), (16), (17) to approach a solution for 0 t tf under assumption 1. The second one with lemma 1 is a mitigation property of the CGM which doesn’t allow to recover high frequencies as well as low frequencies.

Fig. 3.Im (u (t)) = sin ( t) + 2 sin (2 t) + 5 sin (5 t) Observations done on simulations suggest a low contribution of the polynomial part of the solution to the inverse probleme compared to its exponential part. Further investigations should be done with the transfer function [Zwa:04] of (1) to quantify this assumption.

REFERENCES

[Alif:95] O.M. Alifanov, E.A. Artyukhin, S.V. Rumyant- sev, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, begell house, inc., New York, 1995.

[Prud:98] M. Prud’homme and T. Hung Nguyen,On the iterative regularization of inverse heat conduction problems by conjugate gradient method, Int.

Comm. Heat Mass Transfer, vol. 25, n 7, pp.

999-1008

[Zau:06] E. Zauderer, Partial Di¤ erential Equations of Applied Mathematics, Willey, 2006

[Zwa:04] H. Zwart, Transfer functions for in…nite- diemensional systems, Systems Control Letters, vol. 52, n 52(3-4), pp. 247-255, 2004