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VLADIMIR ZUBOV, ALLA ALBU
Dorodnicyn Computing Centre
- f Russian Academy of Sciences
Moscow, Russia
The Use of Fast Automatic Differentiation Technique for Solving - - PowerPoint PPT Presentation
The Use of Fast Automatic Differentiation Technique for Solving - - PowerPoint PPT Presentation
The Use of Fast Automatic Differentiation Technique for Solving Coefficient Inverse Problems VLADIMIR ZUBOV, ALLA ALBU Dorodnicyn Computing Centre of Russian Academy of Sciences Moscow, Russia In studying and modeling heat propagation
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However, another approach is possible: a simplified model is constructed in which the radiative heat transfer is not taken into account, but its effect is modeled by an effective thermal conductivity coefficient that is determined based on experimental data. We consider one possible statement of the inverse coefficient problem. It is considered based on the Dirichlet problem for the
- ne-dimensional unsteady-state heat
- equation. The inverse coefficient problem is
reduced to a variational problem.
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The identification problem of the model parameters
A layer of material of width is considered
L
( )
( , ) ( , ) ( ) 0, ( , ) , T x t T x t C K T x t Q t x x ∂ ∂ ∂ ρ − = ∈ ∂ ∂ ∂
( , 0) ( ), , T x w x x L = ≤ ≤
1 2
(0, ) ( ), ( , ) ( ), . T t w t T L t w t t = = ≤ ≤ Θ
x
- the Cartesian coordinate of the point in the layer
) ( ) ( t L x Q ( , ) T x t
x
t
- the temperature of the material at the point with the coordinate
at the time
ρ
C
and are the density and the heat capacity of the material, respectively
- the coefficient of the convective thermal conductivity
( ) K T
- are given
), (
1 t
w ), (
0 x
w ) (
2 t
w
Direct problem:
) 1 (
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The cost functional :
0 0 2
) , ( ) , ( ) , ( ) (
L
dxdt t x t x t x T T K
2
) ( ) , ( ) , ( t d t t x T t T K
β ≥
- a given number;
( , ) µ ≥ x t
- a given weighting function
( ) t Ρ
- the known heat flux on the left boundary of the domain
( ) K T
( , ) T x t
The optimal control problem is to find the optimal control and the corresponding optimal solution
- f problem (1) that minimizes functional
) (T K
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THE GRADIENT OF FUNCTIONAL IN THE CONTINUOUS CASE
{ }
1 2 [0, ] [0, ] [0, ]
min min ( ), min ( ), min ( )
∈ ∈ Θ ∈ Θ
=
x L t t
a w x w t w t
{ }
1 2 [0, ] [0, ] [0, ]
max max ( ), max ( ), max ( )
∈ ∈ Θ ∈ Θ
=
x L t t
b w x w t w t
Let:
) ( ) ( ) , (
1 2
Q C Q С t x T
{ }
1
( ) : ( ) ([ , ]), ( ) 0, [ , ] = ∈ > ∈ G K z K z C a b K z z a b
- the class of the
feasible control functions
] , [ ) ( ) ( ) (
1
b a C d C z E
z
- the specific internal energy
) (z K
z
- the variation of the control function
) , ( t x T
xt
- the variation of the phase variable
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The Lagrange functional:
0 0
) , ( ) ( ) ( , ) (
L
t d x d x t x T T K x t T E t x p T K I
( )
2 1
, ( ) ( ) p x t C Q C Q
- an arbitrary function
The first variation of
: I
0 0
) , ( ) ( ) ( , ) (
L xt
t d x d x t x T T K x t T E t x p T K I
x t x T t x T K t x ) , ( ) , ( ) , (
Denote:
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The adjoint problem
,
) , ( ) , ( ) , ( 2 ) , ( ) ( ) , ( ) , (
2 2
t x t x T t x x t x p T K t t x p t x T E
, ) , ( Q t x , ) , ( x p
), ( L x
,
) ( ) , ( 2 ) , ( t t t p
, ) , ( t L p
) ( t The first variation of the Lagrange functional
Q z
dt dx T K x t x T x t x p I ) ( ) , ( ) , (
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THE GRADIENT OF FUNCTIONAL
( )
( ) ( ), [ , ] ∇ = Μ ∈ I K T T T a b
, ) , ( ) , ( ) , ( ), , ( ) (
) ( ) (
) ( ) (
i i i i i i
d t J sign x t x p
i Fin i Ini
( , ) ( , ) ( , ) ( , ) ( , )
η
∂ ξ η ∂ ξ η ∂ξ ∂ξ ξ η = ∂ ξ η ∂ ξ η ∂η ∂η
i i i i i
x t J x t x
,
i i
Q Q
,
j i
Q Q
j i
( , ) = ξ η
i
x x
( , ) = ξ η
i
t t
const η =
- the isolines of the field
- f temperature
const ξ =
- the lines orthogonal to
const η =
( )
( ) ( ) Ini Fin
,
i i
η η
- the interval o variation of the isothrems in
i
Q
,
), (
) (
i
Ini
), (
) (i Fin
- the coordinates of the endpoints of the isotherm
contained in
) ( ) ( , i Fin i Ini
const
i
Q
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The interval
] , [ b a
is partitioned by the points
, ~ a T
, ~
1
T
, ~
2
T …,
b TN ~
into N parts. The function
) (T K
to be found is approximated by a continuous piecewise linear functions with the nodes at the points
), ~ ( ~ ~ ) (
1 1 n n n n n n
T T T T k k k T K
, ~ ~
1
n n
T T T
) 1 ,..., ( N n
( ) = %
n n
k K T
N
n n n k
T ) , ~ (
so that THE DISCRETE OPTIMAL CONTROL PROBLEM
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THE DISCRETE OPTIMAL CONTROL PROBLEM ( ) K T
The function was approximated by a continuous piecewise linear function
]) , [ ( b a T
Nonuniform grid :
I i i
x ~
J j j
t ~
) ~ , ~ (
j i j i
t x T T
Finite difference scheme that approximates the direct problem:
i j i j i j i j i j j i i i i i
h T T T K T K T h h C
1 1 1
) ( ) ( ) (
, ) ( ) (
1 1 1 j i i j i j i j i j i j
D h T T T K T K
, 1 , 1 I i
, , 1 J j
i j i j i j i j i j j i i i i i j i
h T T T K T K T h h C D
1 1 1 1 1 1 1 1
) ( ) ( ) 1 ( ) (
1 1 1 1 1 1 1
) ( ) ( ) 1 (
i j i j i j i j i j
h T T T K T K
, ) (
i i
w T
), , ( I i
,
1 j j
w T
,
2 j j I
w T
) , ( J j
1 1
, 0, 1, , 1, ,
− +
= − = − τ = − = % % % %
j j j i i i
h x x i I t t j J
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Cost functional is approximated by the function
with the aid of the trapezoids method:
) ,..., , (
1 N
k k k F F
J j I i j i j i j i j i
h T F T K
1 1 1 2
) ( ) (
J j I i j i j i j i j i
h T F T K
1 1 1 2
) ( ) (
j j j j j j j j j
T T h C T T T K T K h
2 1 1 1 1 1 1 1
2 ) ( ) ( 2 1
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Fast Automatic Differentiation technique
j i j i j i
U Z j i T
, ,
, , ,
) , (
) , ( ) , ( ) , (
, , ,
v q n n
K v q v q v q v q k k n
p U Z v q F k F
v q j i j i
Q v q T v q v q v q T j i
F p U Z v q p
,
, , , ,
, ,
v q j i j i
Z T v q Q
, ,
: ,
v q n j i
U k v q K
, ,
: ,
j i
p
- the values of conjugate variables (impulses)
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Canonical form of the discrete direct problem:
1 1 1 1 1
) ( ) ( ) ( ) (
j i j i j i j i j i j i j i j i j i j i j i j i
T T T T K T K a T T T K T K b T
, ) ( ) ( ) ( ) (
1 1 1 1 1 1 1 1 1 1 1 1 j i j i j i j i j i j i j i j i j i j i j i
T T T K T K d T T T K T K c
) (
1 1
i i i i i j j i
h h h C a
) (
1
i i i i i j j i
h h h C b ) ( ) 1 (
1
i i i i i j j i
h h h C c ) ( ) 1 (
1 1
i i i i i j j i
h h h C d
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The discret adjoint problem
j i j i j i j i j i j i j i
p T T K W T K b p
1 1 1 1
) ( ) (
j i j i j i j i j i j i j i j i j i j i j i
p T T K W T K a T K W T T K b
1 1 1 1
) ( ) ( ) ( ) (
j i j i j i j i j i j i
p T K W T T K a
1 1 1 1
) ( ) (
1 1 1 1 1 1
) ( ) (
j i j i j i j i j i j i
p T T K W T K c
j i j i j i j i j i j i
p T K W T T K a
1 1 1 1
) ( ) (
1 1 1 1 1 1
) ( ) (
j i j i j i j i j i j i
p T T K W T K c
1 1 1 1
) ( ) ( 1
j i j i j i j i j i j i
p T K W T T K c
1 1 1 1
) ( ) (
j i j i j i j i j i j i
p T T K W T K d
j i j i j i j i j i j i j i
T F p T K W T T K d
1 1 1 1 1 1
) ( ) (
1
0, 1, , 0, 1, ,
+ =
= = = =
J j j i I
p i I p p j J
, случае противном в , , ~ ~ если , ~ ~ ) ( ) ( ) (
1 1 1 n j i n n n n n j i j i
T T T T T k k T T T K T K
) ( ) ( ) ( ) (
j i j i j i j i j i
T K T T K T T T T K W
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Gradient of the cost function of the discreet optimal control problem
, ) ( ) ( ) ( ) ( ) ( ) (
1 1 1 1 1
J m I l n m l J j I i j i m l j i J j n j j n j j n
k T K p T K k T K T K F k T K T K F k F
j j j j j j j j
T T h A T T h A T K F T K F
1 1 1 1
2 1 2 ) ( ) (
1
J
A A
1 1 1 1 1 1 1 1
) ( ) ( 2 1 ) ( ) ( 2 2
j j j j j j j j j j
T T T K T K h T T T K T K h A
j j j j
T T h C
2
1
случае, противном в 0, , ~ ~ если , ~ ~ ~ 1 ) (
1 1 n j i n n n n j i n j i
T T T T T T T k T K
случае. противном в 0, , ~ ~ если , ~ ~ ~ ) (
1 1 1 n j i n n n n j i n j i
T T T T T T T k T K
J j ,..., 1
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The value of the gradient of the objective function, calculated according to these formulas is precise for the selected approximation of the
- ptimal control problem.
The machine time needed for calculation the gradient components using the approach presented here (based on the FAD-methodology) is not more than machine time needed for solving
- ne direct problem.
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NUMERICAL RESULTS
1, 1, (0,1) (0,1) = Θ = = × L Q
( , ) ( , ) 1, 1 ρ ≡ ≡ ≤ ≤ T x C T x x
0( )
sin , 1, = ≤ ≤ w x x x
1( )
0, 1 = ≤ ≤ w t t
1.
2( )
sin1exp( 4 ), 1 = − ≤ ≤ w t t t
( , ) sin exp( 4 ), ( , ) x t x t x t Q ϒ = − ∈ 0, sin1 a b = =
4 ) ( T K
1 ) ( x
The first series of computations
- the field functional
SLIDE 19
When approximating the "experimental" field of temperatures by its analytical value
: ) ~ 4 exp( ~ sin ) ~ , ~ (
j i j i j i
t x t x
4
10 403072 . 1
ini
F
26
10 476330 . 3
- pt
F
5
10 201177 . 4 max
ini
GR
17
10 061304 . 6 max
- pt
GR
The maximum deviation of the resulting coefficient of thermal conductivity from its analytical value did not exceed
( ) 4 ≡ K T
5
10 . 1
j j i i
T ϒ =
If
- the solution of the direct problem (1) :
2
10 0196 . 3
ini
F
26
10 9909 . 2
- pt
F
2
10 4852 . 5 max
ini
GR
17
10 2778 . 5 max
- pt
GR
The coefficient of thermal conductivity coincides with accurate to the machine precision
( ) 4 ≡ K T
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N=9
The optimal control
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N=45
The optimal control
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 2 3 4 5 6 7 8 9 10
K(T) T 1
- pt
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 2 2,5 3 3,5 4 4,5 5 5,5
K(T) T 2 3
- pt
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2.
) ( x 1
The functional is the thermal flux on the boundary
1.7408
ini
F
20
10 1.8560
- pt
F
1
10 6.8059 max
ini
GR
14
10 8.5559 max
- pt
GR
The maximum deviation of the resulting coefficient of thermal conductivity from its analytical value did not exceed
( ) 4 ≡ K T
7
10 5 . 3
T T Kini ) (
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1 ) ( x 1
The "mixed" functional
3.
1.7710
ini
F
20
10 6.4531
- pt
F
1
10 7.3544 max
ini
GR
13
10 1.9782 max
- pt
GR
The maximum deviation of the resulting coefficient of thermal conductivity from its analytical value did not exceed
( ) 4 ≡ K T
7
10 . 6
T T Kini ) (
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The second series of computations
0( )
(1.5 ), 1 = − ≤ ≤
m
w x m x x
The input data of the problem coincide with first exemple except for the following ones:
1( )
( 1.5), 1
m
w t m t t = + ≤ ≤
2( )
( 0.5), 1 = + ≤ ≤
m
w t m t t
( , ) ( 1.5 ), ,
m
x t m t x x t Q ϒ = + − ∈
0.5 , 2.5 = =
m m
a m b m > m
- an arbitrary real number
m
T T K ) (
SLIDE 25
- the field functional
1.
1 ) ( x
1 = m
The solution of the identification problem for the coefficient of convective thermal conductivity is not unique
SLIDE 26
- the field functional
2.
1 ) ( x
The solution of the identification problem for the coefficient of convective thermal conductivity is not unique
2 = m
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Necessary conditions for non-uniqueness of solution of the inverse problem
The derivative of the isoline
( ) = x x t
is proportional to the derivative
( , ) ∂ ∂ T x t x
The proportionality coefficient does not change along the isoline and depends only on the temperature
( , ) T x t
- n it .
To single out a unique solution of the optimal control problem, we suggest specifying a point
* T
at which the thermal conductivity coefficient is known:
( ) * * = K K T
) (t x
x t x T t x ) , ( ~ ) ( ,
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If the approximate function
( ) K T
passes through the given point
( , ) * * T K
at each step of the minimization process, then the solution to the inverse problem is unique
0,5 1 1,5 2 2,5 3 2 4 6 8 10 12 T 1 2 3 4
m=1
SLIDE 29
m=2 the flux functional and the "mixed" functional : the approximate values of
( ) K T converged to the limiting function
- pt( ) =
K T T
independently of the initial approximation; the solution was unique
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The third series of computations
the experimental field does not belong to the reachability domain determined by the controls (thermal conductivity coefficients) from the feasible set
( )
{ }
4
( , ) ( , ) ( ) ( ) ( ) ( ) ( , ) 0, ( , ) , (0 1) (0 1) ,
c
x t x t C K q x x t x t Q t x x Q x t ∂ϒ ∂ϒ ∂ ρ ϒ ϒ = ϒ − ϒ = ∈ ∂ ∂ ∂ = < < × < <
( , 0) 1, 1 ϒ = ≤ ≤ x x
(0, ) cos , ( , ) 1, 1 ϒ = ϒ = + ≤ ≤ t t L t t t ( ) 2 3 , ( ) ( ) 1,
c
K C a b ϒ = + ϒ ρ ϒ = ϒ = ≤ ϒ ≤
0( )
0.5, 1 = ≤ ≤ q x x
( ) ϒ
c
K
- the coefficient of the convective thermal conductivity
0( )
= q q x
- a given function
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- 3
10 1.0609
ini
F
5
10 1.1631
- pt
F
- 3
10 2.8158 max
ini
GR
- 10
10 1.4103 max
- pt
GR
1 ) ( x
- the field functional
0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 2,2 0,5 1 1,5 2 2,5 3
K(T) T 1 2 3 4
- pt
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CONCLUSIONS
It is recommended to solve the identification
problem several (3–4) times choosing each time a different function as the initial approximation. If this gives the same solution of the optimal control problem, then it may be considered as the solution of the identification problem. If the solution of the optimal control problem depends on the initial approximation, then an additional condition (e.g., a point at which the thermal conductivity coefficient is known) should be specified in order to solve the identification problem.
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