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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