# Parameter Estimation and Optimum Experimental Design with an by yaofenjin

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```									          Parameter Estimation and
Optimum Experimental Design
with an Adaptive Method of Lines

Dipl. Math. Le Zhang

¨
Diploma Thesis supervised by Dr. Stefan Korkel
BASF Junior Research Group Optimum Experimental Design
Interdisciplinary Center for Scientiﬁc Computing, University of Heidelberg

March 30, 2011              K.U.Leuven

Le Zhang     PE and OED with an Adaptive MOL   –   1
Outline

Problem Deﬁnition and Motivation

Adaptive Method of Lines and Moving Finite Element Method

Numerical Tests

Le Zhang   PE and OED with an Adaptive MOL   –   2
What kind of problem is mainly dealt with?

The time-dependent partial differential equation system:

∂
u(x, t) = L(u(x, t)),      x ∈ Ω ⊂ Rn , t > 0
∂t

subject to appropriate initial and boundary conditions and L is a
differential operator involving spatial derivatives only.

Le Zhang   PE and OED with an Adaptive MOL   –   3
What kind of problem is mainly dealt with?
Example in chemical engineering: time-dependent
convection-diffusion-reaction process:

∂
u(x, t)+   · (v(x, t) u(x, t)) =    · (D(x, t) u(x, t))+f (x, t, u(x, t))
∂t

Convection Term · (v(x, t) u(x, t)): species carried by a ﬂowing
medium with velocity v(x, t)
Diffusion Term · (D(x, t) u(x, t)): effect of diffusion according
to the conservation laws
Reaction Term f (x, t, u(x, t)): consumption and production of
species
Initial and boundary conditions (Dirichlet, Neumann, Robin) must
be imposed to complete the model

Le Zhang    PE and OED with an Adaptive MOL   –   4
Method of Lines

Method of Lines (MOL): one of the most popular and
comprehensive approach to the solution of time-dependent
PDE problems
First discretization in space and then time integration; or
backwards
Typical methods for the spatial discretization, such as ﬁnite
difference, ﬁnite element, ﬁnite volume
Numerical analysis of convergence, stability and
consistency in two steps
Beneﬁt from the successful development of solvers for
ODE/DAE, simple to build in
Most of computer-based PDE solvers nowadays are based
on the idea of MOL

Le Zhang   PE and OED with an Adaptive MOL   –   5
Typical Drawback when using MOL

Fixed spatial discretization unsuitable for problems
exhibiting a high degree of spatial transition
Sufﬁcient spatial discretization nodes are required in order
to guarantee a given numerical accuracy
Waste of grid points in the regions of low spatial activity
Examples: Burger’s equation (ut = −uux + Duxx ),
convection-dominated problem, · · ·

Le Zhang   PE and OED with an Adaptive MOL   –   6

Idea of AMOL: automatic distribution of grid points in order
to locate them to where they are needed
Classiﬁcation of AMOL for PDE with ﬁxed number of grid
points:
Static Regridding: discrete in the time domain, use of
monitor functions to fulﬁll some equidistribution principle,
interpolation of former numerical solutions on the new
mesh after every time step
Dynamic Regridding: continuously in the time domain,
coupling of grid positions and amplitude values, no need of
interpolation, for example: moving ﬁnite element method

Le Zhang   PE and OED with an Adaptive MOL   –   7
Moving Finite Element Method (MFEM)
Basic feature: simultaneous solution of grid positions and
the amplitude values at these positions
For each time point t ∈ [0, T] in spatial interval Ω = [XL , XR ]
the node coordinates:

XL = s0 < s1 (t) < · · · < sN (t) < sN+1 = XR

˜
and set X(t) := (s0 , s1 (t), · · · , sN (t), sN+1 ), then the
approximative solution

U(x, t) =                         ˜
aj (t)αj (x, X(t))
j

where aj (t) < ∞ is the amplitude at the j-th grid point sj (t)
˜
and αj (x, X(t)) the chosen basis functions.

Le Zhang       PE and OED with an Adaptive MOL   –   8
˙
Formulation of U(t)
Differentiate it with respect to t:

N                                           ˜
˙                                ˜              ∂αj (x, X(t)) ˜˙
U(t) =              ˙
aj (t)αj (x, X(t)) + aj (t)               X(t)
˜
∂X
j=1
                                                             
N                                     j+1
                                                ˜
∂αj (x, X(t))         
=             ˙           ˜
aj (t)αj (x, X(t)) + aj (t)                         ˙
si (t)
                                            ∂si              
j=1                                   i=j−1
                           
N                           N             j+1
˜
∂αi (x, X(t)) 
=         ˙            ˜
aj (t)αj (x, X(t)) +       ˙
sj (t)       ai (t)
∂sj
j=1                           j=1             i=j−1

˜
= −Ux αj =: βj (x, X(t))
N                            N
=         ˙            ˜
aj (t)αj (x, X(t)) +          ˙            ˜
sj (t)βj (x, X(t))
j=1                           j=1

Le Zhang         PE and OED with an Adaptive MOL   –   9
Choice of Basis Functions αj and βj


0
               constant on x ∈ (XL , sj−1 ] ∪ [sj+1 , XR ),

at x = sj ,

1

˜
αj (x, X(t)) =   x − sj−1
 sj − sj−1
               linear on x ∈ [sj−1 , sj ],
 sj+1 − x

linear on x ∈ [sj , sj+1 ]

sj+1 − sj


˜                  ˜
βj (x, X(t)) = −Ux αj (x, X(t))

(a)                           (b)

Le Zhang   PE and OED with an Adaptive MOL   –   10
Formulation of a semi-discrete System
Consider the PDE system in one spatial Dimension

ut = L(u),           XL < x < XR , t > 0

subject to appropriate initial and boundary conditions and L is a
differential operator involving spatial derivatives only.
˙
Minimizing the L2 norm of the residual r := U − L(U):

˙
min U − L(U)              2
˙ s
aj ,˙ j                  L2 (Ω)

leads in a semi-discrete system of nonlinear equations:

˙
(r, αj ) = (r, βj ) = 0 ⇔ M(X)X = G(X)

Le Zhang   PE and OED with an Adaptive MOL   –   11
Structure of the Equation System

(αi , αj ) (αi , βj )
M = {Mij }, with block entry Mij =
(βi , αj ) (βi , βj )

M ∈ R2N×2N is symmetric block-tridiagonal and
 
˙
a1                                               
 ˙ 
 s1                                   (L(U), α1 )
˙ 
 a2                                  (L(U), β1 ) 
             
˙ ∈R
 s2     2N                                .
X=                and      G = G(X) =         .        ∈ R2N ,
             
.                                         .
.
             
   .                                (L(U), αN )
˙
aN                                    (L(U), βN )
˙
sN

Le Zhang   PE and OED with an Adaptive MOL   –   12
Regularization with Penalty Functions
For the purpose of avoiding the ill-posedness (singularity of
M), regularization terms are introduced to
                                                 
                    N+1                          
min           ˙
U − L(U) 2 2 (Ω) +    (εj (˙ j − sj−1 ) − Sj )2
s     ˙
L
˙
aj (t),˙ j (t) 
s                                                         
j=1
“internodal viscosity force” εj (˙ j − sj−1 ) penalize relative
s     ˙
motion between nodes and make sure that the system
matrix is positive deﬁnite
“internodal spring force” Sj avoid that neighboring points
move too close with each other
Choice of tuning parameter: a typical way
2
k1                                  2
k2
ε2 =
i                   ,      εi Si =
si − si−1 − δ                  (si − si−1 − δ)2

Le Zhang   PE and OED with an Adaptive MOL   –   13
Simulation Tests with a scalar Reaction-Diffusion
Problem using SolvIND

∂T   ∂2T                        ˜
=     + D (1 + α − T) exp(−δ/T),            t > 0, 0 < x < 1,
∂t   ∂x2
∂T
(0, t) = 0, T(1, t) = 1,          t>0
∂x
T(x, 0) = 1,          0 ≤ x ≤ 1.

T: the reactant temperature
˜     ˜
D = R exp(δ)/(αδ) the Damkohler number with the reaction
rate R
˜
α = 1, δ = 20, R = 5

Le Zhang   PE and OED with an Adaptive MOL   –   14
Simulation Tests with a scalar Reaction-Diffusion
Problem using SolvIND
2                                           2

1.8                                         1.8

1.6                                         1.6

1.4                                         1.4

1.2                                         1.2
MFEM        Solutions        for
1
0   0.2   0.4   0.6         0.8   1
1
0    0.2   0.4    0.6   0.8   1          different    Time         Points
2                                           2
t = 0.26, 0.27, 0.28, 0.29, with
1.8                                         1.8
different choice of tuning
1.6                                         1.6
parameters.
1.4                                         1.4

1.2                                         1.2

1                                           1
0   0.2   0.4   0.6         0.8   1         0    0.2   0.4    0.6   0.8   1

0.3
0.29
0.28
0.27
0.26
0.25
0                       0.2                 0.4        0.6       0.8        1

0.3
0.29
0.28
0.27
0.26
0.25
0                       0.2                 0.4        0.6       0.8        1

0.3
0.29
0.28
0.27
0.26
0.25
0                       0.2                 0.4        0.6       0.8        1

0.3
0.29
0.28
0.27
0.26
0.25
0                       0.2                 0.4        0.6       0.8        1

Le Zhang         PE and OED with an Adaptive MOL   –   15
Simulation Tests with counter-streaming Reactive
Square Waves using SolvIND

Chemical reaction like

A + B → C,
where A and B denote two ﬂuids as reactive species, C is the
product of the reaction. The system of PDEs

∂u1           1 ∂u1
=    −       − ku1 u2
∂t           2 ∂x
∂u2         1 ∂u2
=          − ku1 u2
∂t         2 ∂x

Le Zhang   PE and OED with an Adaptive MOL   –   16
Simulation Tests with counter-streaming Reactive
Square Waves using SolvIND

1                                       1

0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0        20       40      60      80            100
0   20   40   60   80        100

1                                        1

0.8                                      0.8

0.6                                      0.6

0.4                                      0.4

0.2                                      0.2

0                                        0
0   20   40   60   80        100         0        20       40      60      80            100

1                                        1

0.8                                      0.8

0.6                                      0.6

0.4                                      0.4

0.2                                      0.2

0                                        0
0   20   40   60   80        100         0        20       40      60      80            100

1                                        1

0.8                                      0.8

0.6                                      0.6

0.4                                      0.4

0.2                                      0.2

0                                        0
0   20   40   60   80        100         0        20       40      60      80            100

Le Zhang             PE and OED with an Adaptive MOL    –   17
Simulation Tests with counter-streaming Reactive
Square Waves using SolvIND

1

0.8

0.6

0.4

0.2

0
0   20   40   60   80        100

1

0.8                                      140

120
0.6
100
0.4                                      80

60
0.2
40
0                                       20
0   20   40   60   80        100    0

0         20      40      60      80            100

140

1                                       120

100
0.8
80

0.6                                      60

40
0.4
20

0.2                                       0

0         20      40      60      80            100
0
0   20   40   60   80        100

1

0.8

0.6

0.4

0.2

0
0   20   40   60   80        100

Le Zhang             PE and OED with an Adaptive MOL    –   18
Parameter Estimation of a Convection Reaction
Problem using PARFIT

∂u1         ∂u1
= p1     − k u1 u2 ,
∂t         ∂x
∂u2         ∂u2
= p2     − k u1 u2 ,
∂t         ∂x
∂u3
= k u1 u2 .
∂t

with given initial data:
(x − 10.0)2
u1 (x, 0) = exp −                     ,
15.0
u2 (x, 0) = u1 (100 − x, 0),
u3 (x, 0) = 1.0,             for 0 ≤ x ≤ 100.

Le Zhang   PE and OED with an Adaptive MOL   –   19
Parameter Estimation of a Convection Reaction
Problem using PARFIT
Iteration Start: k = 3.0       Iteration Nr. 1: k = 2.999667   Iteration Nr. 2: k = 1.079582
9                                   9                                9
8                                   8                                8
7                                   7                                7
6                                   6                                6
5                                   5                                5
4                                   4                                4
3                                   3                                3
2                                   2                                2
1                                   1                                1
50 60 70 80 90 100 110              50 60 70 80 90 100 110           50 60 70 80 90 100 110

Iteration Nr. 7: k = 1.063547       Iteration Nr. 8: k = 1.158812   Iteration Nr. 9: k = 0.9943803
9                                   9                                9
8                                   8                                8
7                                   7                                7
6                                   6                                6
5                                   5                                5
4                                   4                                4
3                                   3                                3
2                                   2                                2
1                                   1                                1
50 60 70 80 90 100 110              50 60 70 80 90 100 110           50 60 70 80 90 100 110

Iteration Nr. 12: k = 1.001591     Iteration Nr. 13: k = 1.001595          True Value: k = 1.0
9                                   9                                9
8                                   8                                8
7                                   7                                7
6                                   6                                6
5                                   5                                5
4                                   4                                4
3                                   3                                3
2                                   2                                2
1                                   1                                1
50 60 70 80 90 100 110              50 60 70 80 90 100 110           50 60 70 80 90 100 110

The Product Quantity u3 and the Comparation of Measurement Data

Le Zhang              PE and OED with an Adaptive MOL               –   20
Parameter Estimation of a Convection Reaction
Problem using PARFIT
Grid Movement for Iteration Start: k = 3.0
110

100

90

80

70

60

50
0        20               40               60               80      100

Grid Movement for the Last Iteration: k = 1.001595
110

100

90

80

70

60

50
0        20               40               60               80      100

Grid Movement for True Parameter: k = 1.0
110

100

90

80

70

60

50
0        20               40               60               80      100

Change of Grid Movement for different parameter values k

Le Zhang         PE and OED with an Adaptive MOL     –   21
Parameter Estimation of a Convection Diffusion
Reaction Problem using PARFIT

∂y          ∂2y    ∂y
=     D    −v    − r1 yu,
∂t          ∂x2    ∂x
∂z          ∂z
=     v ,
∂t          ∂x
∂u
=     − r2 yu,
∂t

The initial proﬁles are

0.5

(x − 10.0)2
!
0.4                                                           1
y(x, 0) =        exp    −                     ,
0.3                                                           2                  20.0
0.2                                                                                         2
!
1                (x − 40.0)
u(x, 0) =         exp   −                         ,
0.1                                                           10                  25.0
0
0   20   40        60        80     100     z(x, 0) = y(100 − x, 0)

Le Zhang    PE and OED with an Adaptive MOL           –    22
Parameter Estimation of a Convection Diffusion
Reaction Problem using PARFIT

Start of Iterations                     End of Iterations                     True Parameter
1.45                                    1.45                                  1.45

1.3                                     1.3                                   1.3

1.15                                    1.15                                  1.15

1                                       1                                     1

50       70      90        110          50      70      90       110          50     70       90    110

The Comparison of Measured Data and Measurement Data between
the Parameter Values at the First and Last of Parameter Estimation
Iterations and the True Parameters

Le Zhang          PE and OED with an Adaptive MOL     –   23
OED of a Convection Diffusion Reaction Problem
using VPLAN
Let the measurement weight wi for i-th measurement satisfy the
following condition
Nmess
wi ≤ 30,     wi ∈ {0, 1}, Nmess = 96.
i=1

v control variable, start value 0.5, optimal solution 0.4753.

Parameter       Intuitive Exp.           Starting Exp.          Optimal Exp.
D    0.01      +/- 0.00160986           +/- 0.00162366         +/- 0.00124262
r1    1.0      +/- 0.0822818             +/- 0.096872          +/- 0.0596931
r2    0.8      +/- 0.0707931            +/- 0.0779811           +/- 0.052613
A-Criterion      0.00392818               0.00515596             0.00211098
D-Criterion     0.000102418                8.254e-05            7.03587e-05
E-Criterion      0.0115575                0.0152922              0.00617172
M-Criterion      0.0822818                 0.096872               0.0596931

Le Zhang     PE and OED with an Adaptive MOL   –   24
OED of a Convection Diffusion Reaction Problem
using VPLAN
Intuitive Experiment                               Optimal Experiment

1.45                                            1.45

1.3                                             1.3

1.15                                            1.15

1                                            1

50     70           90           110          50          70           90    110

The Measurement Positions and Measurement Data of Intuitive Experiment and Optimal Experiment.

Grid Movement of Intuitive Experiment
110

100

90

80

70

60

50
0               20              40                     60             80          100

Grid Movement of Optimal Experiment
110

100

90

80

70

60

50
0               20              40                     60             80          100

Difference of the Grid Movement between Intuitive Experiment and Optimal Experiment.

Le Zhang                 PE and OED with an Adaptive MOL           –   25