1 adaptive error estimation of the trefftz method for solving the cauchy problem presenter: c.-t....
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1
Adaptive error estimation of the Trefftz method for solving the Cauchy problem
Presenter: C.-T. Chen
Co-author: K.-H. Chen, J.-F. Lee & J.-T. Chen
BEM/MRM 29, 4-6 June 2007, The New Forest, UK
2
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
3
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
4
Numerical methods
Motivation
Mesh methods
Finite Difference methods
Finite Element methods
Boundary Element methods
Meshless methods
Trefftz methods
Method of Fundamental Solution (MFS)(FDM) (FEM) (BEM)
5
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
6
Statement of problem Inverse problems (Kubo) :
1. Lake of the determination of the domain, its boundary, or an unknown inner boundary.
2. Lake of inference of the governing equation.
3. Lake of identification of boundary conditions and/or initial conditions.
4. Lake of determination of the material properties involved.
5. Lake of determination of the forces acting in the domain.
Cauchy problem
7
2 ( ) 0,
u x x D
over-specified condition
1 11
1 1
u ( ) = f ( )B :
t ( ) = g ( )
x x
x x
1 2B = B B
D
22
2
u ( ) = ?
t ( ) =:
?B
x
x
2 ( ) 0,
u x x D
D
1 2B = B B
22
2
u ( ) = ?
t ( ) =:
?B
x
x
1 1
11
u ( ) = f ( )B :
t ( ) = ?x
x x
2 ( ) 0,
u x x D
D
1 2B = B B
22
2
u ( ) = ?
t ( ) =:
?B
x
x
11
1
u ( ) = ?
t ( ) =:
?B
x
x
2 ( ) 0,
u x x D
D
1 2B = B B
22
2
u ( ) = ?
t ( ) =:
?B
x
x
1
11
1
B :t ( ) = g
u ( ) = ?
( )x
x x
Lake of identification of boundary conditions and/or initial conditions.
case 1 case 2
case 3 case 4
8
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
9
Field solution :
where 2N the number of complete functions
jw the unknown coefficient
( )jA x
the T-complete function which satisfies
the Laplace equation
Trefftz method2
1( )( ) j
N
jju x w A x
10
1
1 10 ( , ) () ,, )( n
N N
nn n
nnF Ga bu a
T-complete set functions :
T-complete set
Where: ( , ) cos( )nnF n ( , ) sin( )n
nG n
( ) 1 cos( ) sin( ), =1,, , 2, n njA x n n n
Field solution :
0 j n nw a a b , , The unknown coefficient :
11
Normal differential of the boundary solution
*1
1
*
1
( , )( , ) ( , )( , )
N N
n nx
nn n nat F bu
nG
1 11
1 1
* sincos( ) cos sin(( , ) )
N Nn n
n xn n
F n n n n nr
1 11
1 1
coscos( ) sin sin( )
N Nn n
yn nn n n n n
r
1
1 1
* sinsin( ) cos c( , ) os( )
N Nn n
xn n
nG n n n n nr
1
1 1
cossin( ) sin cos( ) .
N Nn n
yn nn n n n n
r
where
12
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
13
Tikhonov technique
(I)
(II)
2x 2
bAxMinimize
subject to
The proposed problem is equivalent to Minimize
2bAx subject to *
2 x
The Euler-Lagrange equation can be obtained as
bAxIAA TT )(
Where λ is the regularization parameter (Lagrange parameter).
14
The minimization principle xHxb-xAxQxP
2 ][][
in vector notation,
bAxHAA TT )( where
M M M (M-1) (M-1) MH B B
1 -1 0 0 0 0 0 0
-1 2 -1 0 0 0 0 0
0 -1 2 -1 0 0 0 0
0 0 0 0 -1 2 -1 0
0 0 0 0 0 -1 2 -1
0 0 0 0 0 0 -1 1
T
in which
11-000000
011-00000
0000011-0
00000011-
B M1)-(M
Linear regularization method
2[ ]P x A x b where [ ]Q x x H x
15
2 ( ) 0,
u x x D
over-specified condition
x x
x x
1 11
1 1
u ( ) = f ( )B :
t ( ) = g ( )
1 2B = B B
D
22
2
u ( ) = ?
t ( ) =:
?B
x
x
The concept of adaptive error estimation
Step 1:
16
1 2B = B B
D
22
2
u ( ) = ?
t ( ) ==
?B
x
x
2 ( ) = 0,
u x x D
The ill-posed problem
Obtain:
Step 2:
2 2 2 2u ( ) = f ( ), t ( ) = g ( )x x x x
x x
x x
1 11
1 1
u ( ) = f ( )B :
t ( ) = g ( )
By Trefftz method
17
1
11
1
B :t ( ) = g
u ( ) = ?
( )x
x x
1 2B = B B
D
22 2u ( )B : = f ( )x x
2 ( ) = 0,
u x x D
Obtain: 1 1u ( ) = f ( )x x
Step 3:
The well-posed problem
18
The optimal parameterNorm
The solution is more sensitive
The system is distorted
The optimal2
2 2 1Norm : dB 2 2 ufuf2
1 1 1Norm : dB 1 1 ffff
1 : Boundary conditionf 2 : Analytical solutionu
19
Flow chart of adaptive error estimation2. . : ( ) 0,
. . : , ,
G E u x x D
BC
1 1 1 1u ( ) f ( ) t ( ) = g ( )
x x x x x B
2 2u ( ) = f ( )x x
Remedied by the Tikhonov technique Remedied by the Linear Regularization Method
, , Let 2 12 1u ( ) t ( ) = g ( )= f ( ) x x x xx B
1 1u ( ) = f ( )x x
Obtain the left value of the boundary Obtain left the value of the boundary
Obtain the right value of the boundary Obtain the right value of the boundary
obtain the optimal Lamda value obtain the optimal Lamda value T L
End
2 2u ( ) = f ( )x x
2
1: dB 1 11 1f ff fNorm error
1 1u ( ) = f ( )x x
, , Let 2 12 1u ( ) t ( ) = g ( )= f ( ) x x x xx B
2
1: dB 1 11 1f ff fNorm error
Trefftz method
Trefftz method
Trefftz method
Trefftz method
20
Trefftz method for interior problems
Statement of problem
Numerical example
Conclusions
Motivation
Regularization techniques
Outlines
21
over-specified condition
11
1
( ) sin
( ) sin
u x RB
t x
22
2
( ) ?
( ) ?
u xB
t x
2 u( ) 0, x x D
D
Numerical example
‧
R
Circle case:
22
Analytical field solution : ( ) sin , 0 1u x
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
23
Inverse Problem with artificial Contamination
over-specified condition
11
1
( ) sin [1 ]
( ) sin
(1%)ranu x RB
t x
22
2
( ) ?
( ) ?
u xB
t x
2 u( ) 0,x x D
D
‧R
24
1% random errors contaminating the input data
-0.4 -0.2 0 0.2 0.4 0.6 0.8
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
Err
or(
%)
25
Numerical solution without regularization techniques
-0.4 -0.2 0 0.2 0.4 0.6 0.8
-4E+013
-2E+013
0
2E+013
4E+013
6E+013
u(x
)
Analytical solution
Num erical solution
26
Numerical field solution without regularization techniques
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
27
Numerical solutions remedied by 3 different (200 nodes)
( )T TA A I x A b ( H)T TA A x A b
-0.4 -0.2 0 0.2 0.4 0.6 0.8
- 2
0
2
u(x
)
The T ikhonov techn ique(200 nodes)Analytical solution
Num erical solution:
Num erical solution: O p t
Num erical solution:
-0.4 -0.2 0 0.2 0.4 0.6 0.8
- 2
0
2
4
u(x
)
The L inear Regu larization M ethod(200 nodes)Analytical solution
Num erical solution:
Num erical solution: O p t
Num erical solution:
28
(T) 0.0000169
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Numerical field solutions remedied by the Tikhonov technique with 3 different
(T)Opt 0.00169 (T) 0.169
(200 nodes)
29
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(L) 0.0000049 (L) 0.049 (L)Opt 0.00049
Numerical field solutions remedied by the linear regularization method with 3 different
(200 nodes)
30
Obtain the optimal parameters by computing
the Norm deriving from comparing numerical solution with analytic solution
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
100000
1000000
10000000
100000000
1000000000
No
rm
The Tikhonov techniqueNorm with com paring analytical solution
O p t=0.00169
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
100000
1000000
10000000
No
rm
The Linear Regularization MethodNorm with com paring analytical solution
O p t=0.000499
2
2 2 1Norm : dB 2 2 ufuf 2 : Analytical solutionu
31
The norm error of the Tikhonov technique is lower than the linear regularization method
1E-010 1E-007 0.0001 0.1 100
0.01
0.1
1
10
100
1000
10000
100000
1000000
10000000
100000000
1000000000N
orm
Norm of the Tikhonov technique with com paring analytical solutionNorm of the Regularization Method with com paring analytical solution
O p t=0.00169O p t=0.000499
32
The Tikhonov technique and the Linear Regularization Method
-0.4 -0.2 0 0.2 0.4 0.6 0.8
- 2
- 1
0
1
2u
(x)
O p tT =0.00169 ,O p t
L =0.00049 (200 nodes)
Analytical solutionNum erical solution of the Tikhonov techniqueNum erical solution of the Linear Regularization Method
Numerical solutions with optimal (200 nodes)
33
Numerical field solutions with optimal (200 nodes)
The Tikhonov technique The Linear Regularization Method
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(T)Opt 0.00169 (L)
Opt 0.00049
34
Under no exact solution, the optimal results obtained by using the adaptive error estimation
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
100000
1000000
10000000
100000000
1000000000
No
rm
The Tikhonov techniqueNorm with com paring analytical solutionNorm with adaptive error estim ation
O p t=0.00169
O p t=0.00409
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
100000
1000000
10000000
No
rm
The L inear Regu larization M ethodNorm with com paring analytical solutionNorm with adaptive error estim ation
O p t=0.000499
O p t=0.000899
The Tikhonov technique The Linear Regularization Method
35
Trefftz method for interior problems
Statement of problem
Numerical exampleConclusions
Motivation
Regularization techniques
Outlines
36
1. The optimal parameters make the system insensitive to contaminating noise.
2. The present results were well compared with exact solutions.
3. The Tikhonov technique agreed the analytical solution better than another in this example.
4. Under no exact solution, the optimal results are obtained by employing the adaptive error estimation.
Conclusions
38
× The Norm deriving from adaptive error estimation
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
No
rm
Th e L in e a r R e g u la riza tio nN orm w ith adaptive error estim ation
O p t=0.000599
1E-010 1E-007 0.0001 0.1 100
1E-006
1E-005
0.0001
0.001
0.01
0.1
1
Nor
m
Th e Tikh o n o v te ch n iq u eN orm w ith adaptive error estim ation
O p t=0.004
39
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.0002 0.02 Opt 0.002
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Figure 9(a) The numerical field solution remedied by the Tikhonov technique with 3 different lambdas (200nodes)
40
Numerical solution being remedied by the Linear Regularization Method with 3 different lambdaes(200 nodes)
0.000049
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Opt 0.00049
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0.0049
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 9(b) The numerical field solution remedied by the linear regularization method with 3 different lambdas (200nodes)
41
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0 .8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Opt 0.002
×Figure 14 (a), 14(b) Numerical field solution being remedied by the Tikhonov technique and the linear regularization method with optimal lambda (200 nodes)
Opt 0.00049 The Tikhonov technique: The Linear Regularization Method:
42
-0 .4 -0.2 0 0.2 0.4 0.6 0.8
- 2
- 1
0
1
2
u(x)
Th e Tikh o n o v M e th o dAnalytic S olu tion
N um erica l so lu tion(40 nodes)
N um erica l so lu tion(200 nodes)
Numerical solution being remedied by the Tikhonov technique Of 40 nodes and 200 nodes with optimal lambda
43
Numerical solution being remedied by the Linear Regularization Method Of 40 nodes and 200 nodes with optimal lambda
-0 .4 -0.2 0 0.2 0.4 0.6 0.8
- 2
- 1
0
1
2u(x)
Th e L in e a r R e g u la riza tio n M e th o dAnalytic S olu tion
N um erica l so lu tion(40 nodes)
N um erica l so lu tion(200 nodes)