a localized method of particular solutions for solving near singular problems c.s. chen, guangming...
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A Localized Method of Particular Solutions for Solving Near Singular Problems
C.S. Chen, Guangming Yao, D.L. Young
Department of MathematicsUniversity of Southern Mississippi
U.S.A.
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OutlineOutline
• Radial Basis Functions
• The global approaches of the method of particular solutions
• Numerical examples of global method
• Local approach of the method of particular solutions
• Numerical examples of local method
• Near Singular Problems
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Radial Basis Functions
Linear: rCubic: r 3
Multiquadrics: r c c2 2 where is a shape parameter.
Polyharmonic Spines:r r n
r n
n
n
2
2 1
11
log , ,, ,
in 2D, in 3D.
Gaussian: e cr 2
Let : be a continous function with (0) 0. If , letiR R x ,i i i x x x
where is the Euclidean norm. Then is called the RBF.i
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Assume that )(ˆ)( xx ff To approximate f by
f we usually require fitting the given
data set xi
N
1of pairwise distinct centres with the imposed
conditions ˆ( ) ( ), 1 .i if f i N x x
The linear system 1
ˆ ( ) , 1 ,N
i i i ji
f a i N
x x x
is well-posed if the interpolation matrix is non-singular
1i j i NA
x x
Surface Reconstruction Scheme
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The Splitting Method
Consider the following equation
,),( xxfLu( ), ,Bu g x x
Where ,3,2, dRd is a bounded open nonempty domain
with sufficiently regular boundary .Let puuv where pu satisfying )(xfLu p but does not necessary satisfy the boundary condition in (11).
(10)(11)
v satisfies , ,0 xLv. ),()( xxx pugBv
(12)
(13)(14)
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Assume that )(ˆ)( xx ff and that we can obtain an analytical solution up
to
ˆˆ ( ).pLu f xThen
.ˆ pp uu
To approximate f by f we usually require fitting the given
data set xi
N
1 of pairwise distinct centres with the imposed
conditions
.1 ),(ˆ)( Niff xx
Particular Solutions
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The linear system
1
ˆ ( ) , 1 ,N
i ji ii
af i N
x x x
is well-posed if the interpolation matrix is non-singular
1i j i NA
x x
ˆOnce in (*) has been established,f
1
ˆ i
N
p ii
u a
where
i iL and
, .i i i i x x x x
(*)
i iL
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For 2 ,L 2 1( ) d dr dr drr r in 2D
2 2
2 4 41 116 32
1
( ) ln , =
ln ( ) lnd dr dr dr
r r r
r r r r r r r
2 2
2 2 2 2 2 2 231 1
9
( ) +c
( ) ln +c 4 +cc
r r
r c r c r c r
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Where G(r) is the fundamental solution of L
Boundary Method is required.
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The Method of Particular Solutions (MPS)
1
ˆn
p j jj
u u a
,),( xxfLu
( ), ,Bu g x x
j jL where
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Impose boundary conditions
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Once { } is known, the solution of PDEs
can be expressed as followsja
1
ˆ ( )n
j jj
u a r
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Numerical Results
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Example I
Analytical solution:
Computational Domain:
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Consider the Poisson’s equation( , ), ( , )
( , ), ( , )u f x y x y
u g x y x y
Given a large data set 1,
n
i i ix y
( ), ,( ), .
i ii
i i
f x xy
g x x
where
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1x 2x 3x4x 5x
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1x 2x 3x4x 5x
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1x 2x 3x4x 5x
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1x 2x 3x4x 5x
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The absolute errors of LMAPS with L=1, n=5, Sn=100, c=8.9
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L=1, Sn = 100, N=225.
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Local MPS verse Global MPS
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n: number of neighbor points
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LMPS verse LMQ
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Near Singular Problem I
C.S. Chen, G. Kuhn, J. Li, G. Mishuris, Radial basis functions for solving near singular Poisson’s problems,Communication in Numerical Methods in Engineering, 2003, 19, 333-347.
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1.5a
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Profile of exact solution
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CS-RBF
400 quasi-random points
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Test 1 Test 2
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Normalized Shape parameter
where
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Sobel quasi-random nodes Von-Del Corput quasi-random nodes
Random nodes
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Speed up
N=10,000 CPU = 0.5/3.42 sN=40,000 CPU = 3.31/14.06 sN=62,500 CPU = 7.01/25.28 s
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RMSE error verse shape parameter for a=1.6 and various mesh sizes
LMPS
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RMSE error verse shape parameter for h=1/200, and various value of a.
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Near Singular Problem II
Exact solution
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Profile of f(x,y)
f(1,1,) = -15,861, f(0,0)=237
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Near Singular Problem III
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Adaptive Method
First step Second step
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3rd step 4th step
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