desingularized meshless method for solving the cauchy problem speaker: kuo-lun wu coworker :...
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![Page 1: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/1.jpg)
Desingularized meshless method for solving the Cauchy problem
Speaker: Kuo-Lun WuCoworker : Kue-Hong Chen 、 Jeng-Tzong Chen a
nd Jeng-Hong Kao
以去奇異無網格法求解柯西問題
2006/12/16
![Page 2: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/2.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions
![Page 3: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/3.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions
![Page 4: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/4.jpg)
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MotivationNumerical Methods Numerical Methods
Mesh MethodsMesh Methods
Finite Difference Method
Finite Difference Method
Meshless Methods Meshless Methods
Finite Element Method
Finite Element Method
Boundary Element Method
Boundary Element Method
(MFS) (DMM)
![Page 5: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/5.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions
![Page 6: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/6.jpg)
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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 or inputs acting in the domain.
Cauchy problem
![Page 7: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/7.jpg)
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Cauchy problem :
![Page 8: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/8.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for multiple holes Regularization techniques Numerical example Conclusions
![Page 9: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/9.jpg)
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Method of fundamental solutions
Source point Collocation point— Physical boundary-- Off-set boundary
d = off-set distance
d
Double-layer
potential approach
Single-layer
Potential approach
Dirichlet problem
Neumann problem
Dirichlet problem
Neumann problem
Distributed-type
N
jjiji xsUx
1
),()(
N
jjiji xsLx
1
),()(
ijij xsxsU ln),(
s
ijij n
xsUxsT
),(),(
N
jjiji xsTx
1
),()(
N
jjiji xsMx
1
),()(
![Page 10: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/10.jpg)
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The artificial boundary (off-set boundary) distance is debatable.
The diagonal coefficients of influence matrices are singular when the source point coincides the collocation point.
Method of fundamental solutions
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions
![Page 12: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/12.jpg)
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N
1jjij
ON
1jjij
Ii x,sMx,sMx -
Dirichlet problem
Neumann problem
where
N
jjij
ON
jjij
Ii xsTxsTx
1
)(
1
)( ),(),()( Source point Collocation point— Physical boundary
Desingularized meshless method
Double-layer
potential approach
( )
1
( , ) 0,N
Oj i
j
T s x
( )
1
( , ) 0N
Oj i
j
M s x
ixis
1s
2s
3s4s
Ns
I = Inward normal vectorO = Outward normal vector
![Page 13: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/13.jpg)
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1( ) ( ) ( ) ( )
1 1 1
( , ) ( , ) ( , ) ( , ) ,i N N
I I I Ij i j j i j m i i i i i
j j i m
T s x T s x T s x T s x x B
N
1jiij
ON
1jjij
Ii x,sTx,sTx -
1( ) ( ) ( ) ( )
1 1 1
( , ) ( , ) ( , ) ( , )i N N
I I I Oj i j i i i j i j j i i
j j i j
T s x T s x T s x T s x
In a similar way, Bx,x,sMx,sMx,sMx,sMx ii
N
1mii
Iim
IN
1ijjij
I1-i
1jjij
Ii
--
Desingularized meshless method
jixsTxsT
jixsTxsTOi
Oj
Ii
Ij
Oi
Oj
Ii
Ij
),,(),(
),,(),(
( , ) ( , ),
( , ) ( , ),
I I O Oj i j iI I O Oj i j i
M s x M s x i j
M s x M s x i j
![Page 14: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/14.jpg)
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jN
1mNN,mN,N,2N,1
N2,
N
1m2,2m2,2,1
N1,1,2
N
1m1,1m1,
i
MMMM
MMMM
MMMM
)(
)(
)(
--
--
--
j
N
1mNN,mN,N,2N,1
N2,
N
1m2,2m2,2,1
N1,1,2
N
1m1,1m1,
i
TTTT
TTTT
TTTT
-
-
-
Desingularized meshless method
![Page 15: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/15.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation with Cauchy problem Regularization techniques Numerical example Conclusions
![Page 16: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/16.jpg)
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Formulation with Cauchy problem
N Collocation points
M Collocation points
![Page 17: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/17.jpg)
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Derivation of diagonal coefficients of influence matrices.
1)()(2
)(1
12
11 }{][
][
}{
}{
MNMNM
MNN
M
N
A
A
Where ,
N
N
2
1
11}{ ,}{
MN
2N
1N
12
M
,
][
,1
,,3,2,1,
,2,23,21
2,2,21,2
,1,13,12,11
1,1,1
)(1
MNN
MN
mNNmNNNN
MNN
MN
mm
MNN
MN
mm
MNN
aaaaaa
aaaaaa
aaaaaa
A
MN
mMNMNmMNNMNMNMNMN
MNN
MN
mNNmNNNN
MNM
aaaaaa
aaaaaa
A
1,,1,3,2,1,
,11
1,1,13,12,11,1
)(2 ][
,}{
1
2
1
1)(
MN
N
NMN
Formulation with Cauchy problem
![Page 18: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/18.jpg)
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1)()(2
)(1
12
11
][
][
}{
}{
MNMNM
MNN
M
N
B
B
where
,
N
N
2
1
11}{ ,}{ 2
1
12
MN
N
N
M
,
1
2
1
1)(
MN
N
NMN
,
][
,1
,,3,2,1,
,2,23,21
2,2,21,2
,1,13,12,11
1,1,1
)(1
MNN
MN
mNNmNNNN
MNN
MN
mm
MNN
MN
mm
MNN
bbbbbb
bbbbbb
bbbbbb
B
MN
mMNMNmMNNMNMNMNMN
MNN
MN
mNNmNNNN
MNM
bbbbbb
bbbbbb
B
1,,1,3,2,1,
,11
1,1,13,12,11,1
)(2 ][
Formulation with Cauchy problem
![Page 19: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/19.jpg)
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Rearrange the influence matrices together into the linearly algebraic solver system as
1)()(1
)(1
1
11 }{][
][
}{
}{
MNMNN
MNN
N1
N
B
A
The linear equations can be generally written as
bxA
where
,][
][][
)(1
)(1
MNN
MNN
B
AA ,}{ 1)( MNx .
}{
}{}{
11
11
N
Nb
Formulation with Cauchy problem
![Page 20: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/20.jpg)
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Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation with Cauchy problem Regularization techniques Numerical example Conclusions
Outlines
![Page 21: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/21.jpg)
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Regularization technique 1. Truncated singular value decomposition(TSVD)
In the singular value decomposition (SVD), the [A] matrix is decomposed into
TVUA
Where m21 u,,u,uU m21 v,,v,v V and
are column orthonormal matrices,
T denotes the matrix transposition, and
),,,( diag m21
is a diagonal matrix with nonnegative diagonal elements in nonincreasing order, which are the singular values of .
condition number
, Condm
1
1 mwhere is the maximum singular value and is the minimum singular value
ill-condition condition number
![Page 22: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/22.jpg)
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m
2
1
00
00
00
Σ
truncated number then condition number
truncated number = 1
truncated number = 2
Regularization technique 1. Truncated singular value decomposition(TSVD)
![Page 23: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/23.jpg)
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Regularization technique 2. Tikhonov techniques
(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).
![Page 24: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/24.jpg)
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Regularization technique 3. Linear regularization method
The minimization principle xHxb-xAxQxP
2 ][][
in vector notation,
bAxHAA TT )( where
11-000000
1-21-00000
01-21-0000
00001-21-0
000001-21-
0000001-1
BBH M1)-(M1)-(MMT
MM
in which
11-000000
011-00000
0000011-0
00000011-
B M1)-(M
![Page 25: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/25.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical example Conclusions
![Page 26: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/26.jpg)
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Numerical examples
1R
Domain
02
sin
sinRBk
?
?
uB
![Page 27: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/27.jpg)
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The random error
![Page 28: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/28.jpg)
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The boundary potential without regularization techniques
![Page 29: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/29.jpg)
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The boundary potential with different values of λ (or i)
TSVD Tikhonov technique
Linear regulariztion method
![Page 30: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/30.jpg)
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L2 norm by different regularization techniques
TSVD Tikhonov technique
Linear regulariztion method
![Page 31: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/31.jpg)
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The boundary potential with the optimal value of λ (or i)
TSVD Tikhonov technique
Linear regulariztion method
![Page 32: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/32.jpg)
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The boundary potential with the optimal value of λ (or i)
![Page 33: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/33.jpg)
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L2 norm by different regularization techniques
![Page 34: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/34.jpg)
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Outlines Motivation Statement of problem Method of fundamental solutions Desingularized meshless method Formulation for Cauchy problem Regularization techniques Numerical examples Conclusions
![Page 35: Desingularized meshless method for solving the Cauchy problem Speaker: Kuo-Lun Wu Coworker : Kue-Hong Chen 、 Jeng-Tzong Chen and Jeng-Hong Kao 以去奇異無網格法求解柯西問題](https://reader036.vdocuments.net/reader036/viewer/2022081501/56649d7f5503460f94a62fcd/html5/thumbnails/35.jpg)
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Conclusions
Only selection of boundary nodes on the real boundary are required.
Singularity of kernels is desingularized. Linear regularization method agreed the an
alytical solution better than others in this example.
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The end
Thanks for your attentions.
Your comment is much appreciated.