c gasparadvances in numerical algorithms, graz, 2003 1 fast interpolation techniques and meshless...

1Advances in Numerical Algorithms, Graz, 2003

C Gaspar

Fast interpolation techniques and Fast interpolation techniques and meshless methodsmeshless methods

Csaba GáspárSzéchenyi István University,

Department of Mathematics

Győr, Hungary

[email protected]

C Gaspar Advances in Numerical Algorithms, Graz, 2003

2

• surface approximation

• completion of incomplete data sets

• construction of meshfree methods

4: ESWN

Cffff

f

0f

MotivationsMotivations


3

The scattered data interpolation problem inThe scattered data interpolation problem in 2R

Vectorial problems: some additional conditions are required

0 div u

0rot u

),...,1()(

such that :function smooth)tly (sufficien a Find

.,..., values theand

in spaced)ly (irregular ,..., points Given the :

2

1

21

Nkuxu

u

uu

xxProblem

kk

N

N

RR

R


4

Method of radial basis functionsMethod of radial basis functions

where the basis functions depend on ||x|| only.The coefficients are a priori unknown and can be determined by solving the interpolation equations:

The form of the interpolation function:

N

jjjj xxxu

1)(:)(

m

jjj xpa

1)(

),...,2,1()( Nkuxu kk

and the orthogonality conditions:

),...,2,1(0)(1

mkxpaN

jjkk


5

Some popular RBFsSome popular RBFs

Method of Multiquadrics (Hardy):

Thin Plate Splines (Duchon):(biharmonic fundamental solution)

Traditional DRM (Partridge, Brebbia):

22:)( jj crr

rrrj log:)( 2

Gaussian functions:22

:)(rc

jjer

rrj 1:)(

Compactly supported RBFsCompactly supported RBFs (Wendland):

.........

)14()1(:)(

)1(:)(

4

2

rcrcr

rcr

jjj

jj

Globally supported RBFsGlobally supported RBFs


6

Why do we like the Why do we like the RBFs soRBFs so much? much?

Main advantages:

• excellent approximation properties

• easy to implement

But what is the But what is the price to payprice to pay??

We have to face that:

• RBFs generally lead to a system with large, dense and ill-conditioned matrix

• memory requirement: O(N 2)

• computational cost: O(N 3)


7

• domain decomposition• preconditioning• fast multipole evaluation methods• compactly supported RBFs• multi-level techniques• direct multi-elliptic interpolationdirect multi-elliptic interpolation

RemediesRemedies


8

Methods based on iterated elliptic operatorsMethods based on iterated elliptic operators

Observation: Thin plate splines are fundamental solutions of the biharmonic operator

Idea: Let us define the interpolation function in such a form:

)conditionstion (interpola :)( kk uxu )conditions(boundary 0:| u

• Second-order operators: unusable!• Fourth-order operators:

biharmonic operator; bi-Helmholtz-operator; Laplace-Helmholtz-operator

• Higher order operators:triharmonic operator; tetraharmonic operator; iterated Helmholtz-operator, etc.

},...,,{\in 0 21 NxxxLu


9

””Harmonic interpolation”: a counterexampleHarmonic interpolation”: a counterexample

),...,2,1(),(:),(

0:|

)},(),...,.(),,{(\in 0

0

2211

Nkyxfyxu

u

yxyxyxu

kkkk

NN

Numerical features:•ill-posed problem•poor interpolation surface•logarithmic singularities logarithmic singularities at the interpolation pointsat the interpolation points

yxyxf sinsin:),( ),1,0()1,0( : 0


10

Biharmonic interpolationBiharmonic interpolation

}0)()(:)({: 120 NxwxwHwW

)(20 Hu

),...,2,1()()(

},...,,{\in 0

0

21

Nkxuxu

xxxu

kk

N

Equivalent formulations: Let us introduce the closed subspace:

Classical formulation: For a given function , find a function such that

Direct problem: For a given function , find a functionsuch that

)(200 Hu

},...,,{\in 0)( 210 Nxxxvu

Variational problem: For a given function , find a function such that

)(200 Hu

)(200 Hu

Wv

Wv Wwwvu L 0),( )(0 2

Theorem: The direct and the variational problems are equivalent and have a unique solution in W.


11

Representation of the biharmonic interpolationRepresentation of the biharmonic interpolation

Theorem: u is uniquely represented in the form:

where w is an everywhere biharmonic function, and is the biharmonic fundamental solution:

N

jjj xxxwxu

1)()()(

rrr log)( 2

...but the coefficients can be calculated from: without solving any system of equations!without solving any system of equations!

This is an RBF-like interpolation...This is an RBF-like interpolation...

N

jjj xxxu

1)()(


12

ApproximationApproximation

||||infsup:},...,{ 1

yxhNxxyx

Separation distance:

Theorem: The following estimations hold:

)(02)(0

)(02

1)(0

20

10

202

||||||||

||||||||

HH

HL

uhCuu

uhCuu


13

Bi-Helmholtz interpolationBi-Helmholtz interpolation

)(20 Hu

),...,2,1()()(

},...,,{\in 0)(

0

2122

Nkxuxu

xxxuIc

kk

N

Classical formulation: For a given function , find a function such that

Direct problem: For a given function , find a functionsuch that

)(200 Hu

},...,,{\in 0)()( 21022

NxxxvuIc

Variational problem: For a given function , find a function such that

)(200 Hu

)(200 Hu

Wv

Wv Wwwvu L 0),( )(0 2

The direct and the variational problems are equivalent, and have a unique solution in W. The representation and approximation theorems are still valid.

Now the fundamental solution is Now the fundamental solution is rapidly decreasingrapidly decreasing ("almost compactly supported").("almost compactly supported").

Representation formula:

Coefficients can be computed directly:

)()(,)()( 11

crrKrxxxuN

jjj

N

jjj xxxu

1)()(


14

Further generalizationsFurther generalizations

• The use of even higher order differential operators• The use of different operators simultaneously

In each case, the representation automatically produces an RBF-likeIn each case, the representation automatically produces an RBF-likeinterpolation based on the fundamental solution of the applied operator.interpolation based on the fundamental solution of the applied operator.

0uTriharmonic operator:

Mixed Laplace-Helmholtz operators:

0)( 2 fIc

0)( 2 fIc

0)( 22 fIc


15

The main computational problem:The main computational problem:

How to solve How to solve the multi-elliptic interpolation equation?!the multi-elliptic interpolation equation?!


16

Numerical techniquesNumerical techniques

Main features:• No algebraic interpolation equationNo algebraic interpolation equation has to be solved.• Instead, the problem is converted to an auxiliary differential equationauxiliary differential equation.• The domain is not predefinedThe domain is not predefined• The boundary conditions are not strictly prescribedThe boundary conditions are not strictly prescribed

Numerical solution techniques:• Traditional FDM or FEM ?• FFT ?• MGR ?• quadtree-based multigrid techniquesquadtree-based multigrid techniques


17

Quadtree cell systemsQuadtree cell systems

• Non-uniform cell subdivision generated by a finite set of points• Schemes based on

finite volumes• Multigrid solution

technique• Computational costComputational cost

can be reduced to can be reduced to

OO((NN log log NN))

Demo application


18

BBiharmonic iharmonic interpolation on a QT cell systeminterpolation on a QT cell system::an ean examplexample

: ( , ) ( , ), ( , ): sin sin 0 1 0 1 2 20f x y x y

The QT-cell system The interpolation functionRelative L2-errors (%):

N \ lmax 7 8 9 10 32 17.314 7.759 8.305 9.202 64 4.159 4.026 4.529 2.692128 0.770 1.332 0.691 0.581256 0.213 0.154 0.143 0.148512 0.095 0.062 0.049 0.041


19

Bi-Helmholtz interpolation: an exampleBi-Helmholtz interpolation: an example

Relative L2 -errors (N := 200):

c 10 15 20 25 30 40 50 Error 1.55 1.63 2.31 4.24 7.20 15.22 25.01

))4

1(2(sin))

4

1(2(sin:),(),1,0()1,0(: 22

0 yxyxu

c = 10

c = 50


20

Construction of meshless methods.Construction of meshless methods.General approachesGeneral approaches

Original problem: )(in fLu + boundary conditions

Method 1. The solution u is directly interpolated:

Method 2. First, the function f is interpolated:

M

jjj xxxu

1)(:)(

M

jjj xxxf

1)(:)(

M

jkjkj fxxL

1)(

LxxxuM

jjkj with)(:)(

1


21

Construction of meshless methodsConstruction of meshless methodsby direct multi-elliptic interpolationby direct multi-elliptic interpolation

Model problem: )(in fu + boundary conditions

Method 1. The solution u is directly interpolated by tetraharmonic interpolation:

Method 2Method 2. First, the function . First, the function ff is interpolated by biharmonic interpolation. is interpolated by biharmonic interpolation. Then, Then, using the same QT cell systemusing the same QT cell system, the model problem is, the model problem is solved, solved, without boundary conditionswithout boundary conditions..

),...,1()(

),...,2,1()(

},...,,{\in 0 214

NMkuxu

Mkfxu

xxxu

kk

kk

N

...A bit uncomfortable......A bit uncomfortable...

This procedure mimics the computation of Newtonian potentialThis procedure mimics the computation of Newtonian potentialReduces the domain problem to a Reduces the domain problem to a boundary problemboundary problem

}0)(...)(

,0)(...)(:)({:

1

140

NM

M

xuxu

xuxuHwW


22

How to solve the boundary problem?How to solve the boundary problem?

Traditional tool: the Boundary Element Method

Direct Multi-Elliptic Boundary InterpolationDirect Multi-Elliptic Boundary Interpolation

DrawbacksDrawbacks:• large, dense and ill-conditioned matrices

AdvantagesAdvantages:• reduction of dimension• reduction of mesh


23

Boundary interpolation 1. The basic ideaBoundary interpolation 1. The basic idea

Model problem: Laplace equation with Dirichlet boundary condition

The interpolation points are located at the boundary only, and...The interpolation points are located at the boundary only, and...

The use of the mmixed Laplace-Helmholtz operatorixed Laplace-Helmholtz operator:

0u'Harmonic' interpolation:

0uBiharmonic interpolation:

'Quasi-harmonic' interpolation:

0)( 2 uIc

0)( 2 uIc

02

1( ) ( ( || ||) log( || ||))

2x K c x c x

c Fundamental solution:


24

Boundary interpolation 2. Boundary interpolation 2. Application to the Laplace equationApplication to the Laplace equation

Key issue: the proper choice of the scaling constant c

The original problem: 0* 0, * |u u u

The associated boundary interpolation problem:

002|,|,0

1v

n

uuuu

cu

Theorem 1: 1/ 2 1/ 22 ( ) 0 0( ) ( )|| * || (|| || || || )L H H

Cu u u v

c

1 12

0 0( ) ( ) ( )|| * || (|| || || || )

H H L

Cu u u v

c Theorem 2:

With more regular data:


25

Dirichlet boundary conditionDirichlet boundary conditionExamplesExamples

%)718.1(500c

%)149.0(250c

%)102.26(0c


26

Neumann boundary conditionNeumann boundary conditionThe naive approachThe naive approach

The original problem:


),...,2,1()()(

},...,{\in ,0)(

0

122

Nkxvxn

u

xxuIc

kk

N

0|*

,0* vn

uu

)(30 Hu

The method fails...The method fails...

uu approximates approximates vv00 but fails to approximate but fails to approximate uu* !* !


27

Neumann boundary conditionNeumann boundary conditionAn improved approachAn improved approach

Idea: Locate additional Dirichlet points along Neumann boundary in outward normal direction; Define Dirichlet data to enforce Neumann conditions

0:~ vduu

)(:~0 n

uvduu

or:

This procedure mimics theThis procedure mimics themethod of fundamental solutionsmethod of fundamental solutions


28

Neumann boundary conditionNeumann boundary conditionExamplesExamples

025.0d

050.0d

100.0d


29

Boundary interpolation 3.Boundary interpolation 3.Application to the biharmonic equationApplication to the biharmonic equation

The original problem: 0 0*

* 0, * | ,u

u u u vn


2

0 0 02 2

10, | , ,

u uu u u u v w

nc n

Theorem 3: 1 3/ 2 1/ 2 1/ 20 0 0( ) ( ) ( ) ( )|| * || (|| || || || || || )

H H H H

Cu u u v w

c

Further applications:• 2D Stokes flow


30

Vectorial interpolation problemsVectorial interpolation problems

),...,1()(such that :function a Find

.,..., values theand in ,..., points Given the :22

21

21

Nkx

xxProblem

kk

NN

uuRRu

RuuR

Componentwise interpolation works well...

... but often additional conditions are required:

0 div

or ,0rot

u

u

Use the potential/stream function approach

RBF-method failsRBF-method fails

),...,2,1()(,)( 21 NkuxDvxD kkkk


31

Tri-Helmholtz-interpolation for vectorial Tri-Helmholtz-interpolation for vectorial problemsproblems

Classical formulation: For a given functionfind a function such that

),...,2,1()()(),()( 21 NkxuxDxvxD kkkk

Variational formulation: DefineFor a given function find a function such that

},...,1,)( grad :)({: 30 NkxwHwW k 0

)()(),( 22 HHvuu

)(300 H

)(30 H

)(30 H

WwwIcIcc

wIcIc

L

L

0)(),)((

)(grad),)((grad

)(

20

22

)(

20

2

2

2

Theorem: The variational problem has a unique solution in W.

The vectorial interpolation function: (grad ) is divergence-free.

yxyx 2sin2sin),(


32

Representation and RBF-like solutionRepresentation and RBF-like solution

Representation formula: If c > 0 then is represented in the form:

where is the fundamental solution of the operator :

The theory assures existence and uniqueness also for the The theory assures existence and uniqueness also for the RBF-like interpolationRBF-like interpolation:

32 )( Ic

N

jjj

N

jjj xxDxxDx

12

11 )()()(

),...,2,1()()(

)()(

122

112

112

111

NkuxxDxxD

vxxDxxD

k

N

jjkj

N

jjkj

k

N

jjkj

N

jjkj

))(2)((16

1)( 10

224

rcrcKrcKcrc

r

Componentwise TPSExactVectorial tri-Helmholtz


33

Summary and conclusionsSummary and conclusions

• Interpolation based on iterated elliptic equationsInterpolation based on iterated elliptic equations• Boundary interpolation by singularly perturbed Boundary interpolation by singularly perturbed equationsequations• Special RBF- and RBF-like methods are Special RBF- and RBF-like methods are automaticallyautomatically generatedgenerated

• No explicit use of RBFs is neededNo explicit use of RBFs is needed• No large and dense algebraic equations are to be solvedNo large and dense algebraic equations are to be solved• Quadtree-based multigrid methods are usedQuadtree-based multigrid methods are used

34Advances in Numerical Algorithms, Graz, 2003

C Gaspar

ThankThank you you for your for your attention!attention!

c gasparadvances in numerical algorithms, graz, 2003 1 fast interpolation techniques and meshless...

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