Docstoc

Parameter Estimation and Optimum Experimental Design with an

Document Sample
Parameter Estimation and Optimum Experimental Design with an Powered By Docstoc
					          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
Adaptive Method of Lines (AMOL)


     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
Thank you for your attention!




          Le Zhang   PE and OED with an Adaptive MOL   –   26

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:15
posted:8/26/2011
language:English
pages:26