Funciones Radiales y M´etodos Adaptivos Pedro Gonz´alez ... › smc-2007 › Archivos ›...

Descomposicion de Dominio conFunciones Radiales y Metodos Adaptivos

Pedro Gonzalez-Casanova Henrıquez

DGSCA, UNAM

Jose A. Munoz Gomez

Gustavo Rodrıguez Gomez

INAOE

Seminario de Modelacion Matematica y

Computacional

Instituto de Geofısisca, UNAM, Mexico

2007

Introduction

Why RBF-PDE methods?

Introduction


• Classical methods like finite differences, finite

elements, etc. are based on grids.

Introduction




• In 2D frequently, 70 % of CPU time of real life

PDE’s problems are used on grid generation.

Introduction






• In 3D, mesh generation is even more time con-

suming and moving mesh methods are extremely

expensive.

Introduction






• In 3D, mesh generation is even more time con-

suming and moving mesh methods are extremely

expensive.

• Physics is modified according to the numerical

method and not the way around.

Introduction

• Radial basis functions: φ : IR+ → IR, ‖ ‖ Euclidian

norm of IRd

Φ(x) ≡ φ(‖ ‖) : IRd → IR

• x1, ..., xN ∈ Ω ⊂ IRd, set of nodes. RBF inter-

polant:

s(x, ε) =N∑

j=1

λiφ(‖x − xj‖, ε) + pn(x)

• Interpolation conditions:

s(xi, ε) = u(xi), i = 1, ..., N

Introduction

• No grid is needed to built the RBF interpolant.

Introduction


• The fact that the RBF depends on the Eu-

clidian norm, implies the Gramm matrix has the

same complexity in 2D and 3D.

Introduction





• Partial derivatives can be approximated in a

highly accurate way by using RBF.

Introduction







• To solve PDE, several methods have appeared:

Introduction







• To solve PDE, several methods have appeared:

– Colocation techniques, (un-symmetric and

symetric)

– Moving last square methods with RBF ker-

nels

The problem

RBF collocation methods suffer from a bad condi-

tioning as the number of nodes increases. Strate-

gies to face this problem:

1. Domain decomposition methods.

The problem





2. Node and shape parameter adaptive algorithms,

and

The problem






and

3. Preconditioning thechniques.

The problem






and

3. Preconditioning thechniques.

In this talk we shall be concerned with points (1)

and (2).

Outline of the talk

Up to our knowledge, no algorithm has appeared

which combines Domain decomposition techniques

with node adaptive methods. In this talk:

Outline of the talk




1. Review Domain decomposition methods.

Outline of the talk





2. Formulate a Node and shape parameter addaptive

algorithms, and

Outline of the talk






algorithms, and

3. Formulate a new algorithm which combines these

techniques .

Outline of the talk






algorithms, and

3. Formulate a new algorithm which combines these

techniques .

More precisely, we have the following objective.

Objective:

• Build a numerical scheme based on sets of non

uniform node distributions, rather than on Carte-

sian grids.

Objective:



sian grids.

• Build a node adaptive thinning RBF method for

PDE’s based on scattered data.

Objective:



sian grids.



• Combine a domain decomposition algorithm with

the node adaptive thinning technique to solve the

PDEs problems.

Objective:



sian grids.



• Combine a domain decomposition algorithm with

the node adaptive thinning technique to solve the

PDEs problems.

• Numerical examples will be presented for a Pois-

son and a stationary convection diffusion prob-

lems in 2D.

RBF Domain Decomposition Methods

Several Domain Decomposition Methods for RBFs

collocation techniques have been recently formu-

lated:

• For stationary PDE’s, [5,18,7,19,8,10,9,13].




lated:


• Evolutionary PDE’s, [15, 3].




lated:



• In Zhang et. al. [21], a convergence prove is

given for Schwarz additive and multiplicative

methods.




lated:



• In Zhang et. al. [21], a convergence prove is

given for Schwarz additive and multiplicative

methods.

These methods have ben formulated in 2D, both

for linear and non linear problems.


Dispite these considerable advances, several lim-

itations and open problems remains to be solved

• Except for Wong et al. [15], which uses a

multizone formulation for a non-regular node

distribution, all these works has been formulated

for Cartesian Grids.








• None of the former algorithms do combine adap-

tive techniques with DD methods.








• None of the former algorithms do combine adap-

tive techniques with DD methods.

• Moreover, the number of sub domains remains

fixed during the iteration process.

Schwarz Methods for RBF-PDE

All these methods are based on Schwarz approach,

Zhou et. al. [19]:



Zhou et. al. [19]:

• Schwarz additive method: Easer to parallelize

than the multiplicative version.



Zhou et. al. [19]:



• Schwarz multiplicative method. Twice faster

than the additive version.



Zhou et. al. [19]:



• Schwarz multiplicative method. Twice faster

than the additive version.

In this talk we shall consider the multiplicative

Schwarz method in the context of a non uniform

data distribution.


DD-RBF Schwarz Ovelapping Methods. Generic el-

liptic PDE problem:

Lu = f in Ω ⊂ IRd,

Bu = g on ∂Ω,

In Kansa’s approach, the exact solution u(x) of

the problem is approximated by the expansion

u(x) =N∑

j=1

λjφ(‖x − xj‖) + p(x), x ∈ IRd,

where p ∈ πdm is a polynomial of degree at most

m in d-dimensions, which depend on the RBF.


Some classical RBF:

Multiquadric (MQ) φ(r) =√

r2 + c2

Inverse Multiquadric (IMQ) φ(r) = 1/√

r2 + c2

Gaussian (GA) φ(r) = e−(cr2)

Thin-Plate Splines (TPS) φ(r) = rmlogr, m = 2, 4, 6, . . .Smooth Splines (SS) φ(r) = rm, m = 1, 3, 4, . . .

The collocation technique applied to the former

problem, gives us the linear system:

AΛ = F,


where

• F = [f g]T , A = [LΦa BΦb]T ,

– Λ ∈ IRN ,

– Φa ∈ IRni×N ,

– Φb ∈ IRnf×N , N = ni + nf .

• The entries of the matrix A are

– LΦa = Lφ(‖xi − xj‖), i = 1, . . . , ni and

– BΦb = Bφ(‖xi − xj‖), i = ni + 1, . . . , N ,

j = 1, . . . , N .

RBF Schwarz Methods: Artificial boundaries are updatedfrom the boundary vector F i.

Λn+1i = Λn

i + A−1i (F i − AiΛ

ni ), n = 0, 1, 2, ...

BUT ui = HiΛi i = 1, 2, where Hi = (φ1, ..., φni)

un+1i = un

i + HiA−1i (F i − AiH

−1i un

i ), n = 0, 1, 2, ...

for i = 1, 2.


• RBF additive version: Artificial boundaries are

updated from the results of the nth step.

un+1i = un

i + HiA−1i (F n

i − AiH−1i un

i ), n = 0, 1, 2, ...

for i = 1, 2.

• RBF multiplicative version: Artificial boundaries

updated from most recent neighboring subdo-

mains. For n = 0, 1, 2, ...

un+1/21 = un

i + H1A−11 (F n

1 − A1H−11 un

1 )

un+12 = u

n+1/22 + H2A

−12 (F

n+1/22 − A2H

−12 u

n+1/22 ),

Adaptive RBF Collocation Method

This algorithm combines Behrens et al. refine-

ment method with a quad-tree type technique.

• Use unsymmetric collocation to solve the PDE.





• Let Ξ be the initial set of nodes. For each node

x ∈ Ξ select a set of neighboring nodes Nx\x ∈ Ξ.







• Use a quadtree type technique to locate the neigh-

boring nodes.







• Use a quadtree type technique to locate the neigh-

boring nodes.

• Construct a local TPS interpolant r2i log(ri)+p1:

Iu(x) based on Nx\x


• Let u(x) the RBF numerical approximation of

PDE and Iu(x) the local interpolant. Define

the error indicator

η(x) = |u(x) − Iu(x)|, (1)




the error indicator

η(x) = |u(x) − Iu(x)|, (1)

• For all x If η(x) > θr refine. If η(x) < θc remove.




the error indicator

η(x) = |u(x) − Iu(x)|, (1)

• For all x If η(x) > θr refine. If η(x) < θc remove.

• Node insertion/remotion is performed by using

the quadtree algorithm. For each cell marked

to be refined, four nodes are inserted, following

the rule: 2:1.

RBF Adaptive Methods

• Behrens et al. method uses a Delauny triangu-

lation, our approach a quadtree technique.




• Quadtree technique can be replaced by octree

for 3D problems, thus extending the algorithm.






• In 1D several adaptive methods have appeared

in the literature.






• In 1D several adaptive methods have appeared

in the literature.

In 2D and for scattered data we find the following

approaches:


• The method presented in this talk is based on

a local error estimate and a quadtree algorithm in

order to perform the knot insertion/remotion

of nodes.





of nodes.

• Driscoll et al., [4] formulated a method based

on a global error estimate and a quadtree tech-

nique to perform the adaptive knot strategy to

build their algorithm.





of nodes.

• Driscoll et al., [4] formulated a method based

on a global error estimate and a quadtree tech-

nique to perform the adaptive knot strategy to

build their algorithm.

• Moreover, in Pereyra et al., [20] a global op-

timization method is formulated to solve the

PDE, which adapts the nodes location and the

shape parameters simultaneously.

Local node refinement and DomainDecomposition Algorithm:

• The following algorithm, is based on a Multi-

quadric collocation method.




• Numerical experiments with thin plate splines

r4log(r) where performed and indicated that

multiquadrics are faster for a given error.







• The local error η(x) is define by a thin plate

spline RBF: r2log(r) + x + y + 1.







• The local error η(x) is define by a thin plate

spline RBF: r2log(r) + x + y + 1.

• An initial value of the shape parameter cini is

given, and it is actualized at each iteration by

cnewj = cold

j /2.


• A recursive coordinate bisecction is use to form

the subdomains.



the subdomains.

• A given number Nmax is used to determine

whether the subdomain should be divided or not.



the subdomains.



• The number levelini determines whether a given

node is refined or not: a node can be removed

only if its level in the tree is greater than levelini



the subdomains.



• The number levelini determines whether a given

node is refined or not: a node can be removed

only if its level in the tree is greater than levelini

• The stopping criteria for the Schuarz multiplica-

tive algorithm is given by: ‖uk − uk−1‖ ≤ 0.001,

where k denotes the iteration of the DDM

method.


1. Define The variables: levelini, cini, θr, θc, Nmax.



2. Built the initial tree and refine two times in the

boundary




boundary

3. While maxη(x) > θr or iterations <50




boundary


4. Partition Ω by recursive coordinate bisecction

and form the overlaping zones.




boundary




5. Solve PDE by multiplicative Schwarz algorithm.




boundary





6. Join the solution obtained in each Ωi to built the

soution in Ω.




boundary





6. Join the solution obtained in each Ωi to built the

soution in Ω.

7. Perform the local refinement of nodes. End While.


1. The stopping criteria: there are no more nodes

to insert or number of iterations exceeded.


1. The stopping criteria: there are no more nodes

to insert or number of iterations exceeded.

2. Overlapping zones are built by using the quadtree

structure: The more level on the tree the more

traslapping nodes.

RefinementLevel Expansion5 - 6 26 - 10 311 - 14 4

Moreover: the more refinement level, the greater

gradient of the solution.


1. To join the solution of each sub-domain, a node

is index both to the domain and to the sub-

domain.




domain.

2. Given a sub-domain Ωi: if each node of all other

sub-domains do not belongs to Ωi then it is in-

cluded in Ωi and otherwise discarded.




domain.

2. Given a sub-domain Ωi: if each node of all other

sub-domains do not belongs to Ωi then it is in-

cluded in Ωi and otherwise discarded.

3. Point (7) corresponds to Schwarz and local thin-

ing perviously seen.

Numerical Results:

Poisson Problem:

∂2u

∂x2+

∂2u

∂y2= f en Ω ⊂ R2, (2)

Convection Diffusion Problem:

β∇2u + v · ∇u = 0 x ∈ Ω ⊂ IR2, (3)

in both problems Dirichlet boundary conditions are

considered.

Poisson Equation: As iteration increases: error tends to zeroand the number of domains increases

Iter Party N e_rms e_inf0 2 2152 2.669998924940 13.3818063379121 2 2884 0.764945619834 3.0789971096302 4 3589 0.034990848404 0.1083082937013 4 5344 0.003880880328 0.0105377659304 8 7252 0.003988968225 0.0102399888965 8 8548 0.002610737155 0.0068887014006 8 8686 0.002520146497 0.0068920291977 8 8728 0.002505314335 0.006890463882

Error in each sub-domain:

[0] rms = 0.0025494 e_inf = 0.006890[1] rms = 0.0025529 e_inf = 0.003909[2] rms = 0.0025491 e_inf = 0.006890[3] rms = 0.0027670 e_inf = 0.004143[4] rms = 0.0025112 e_inf = 0.003887[5] rms = 0.0023968 e_inf = 0.006883[6] rms = 0.0023192 e_inf = 0.003637[7] rms = 0.0023970 e_inf = 0.006883

Convection Diffusion Equation: As iteration increases: errortends to zero and the number of domains increases. Pe = 1000.0

Iter Party N e_rms e_inf0 2 2152 0.055968003702 0.3647553841261 5 4714 0.006318200112 0.0824869612412 5 4909 0.008948974295 0.0811974179723 10 7609 0.012955619297 0.0526049472264 10 7723 0.012924100339 0.0504036994175 10 7744 0.012912038660 0.0504261940576 10 7747 0.012910525584 0.050441756859

Error in each sub-domain:[0] rms = 0.0182151 e_inf = 0.050442[1] rms = 0.0161961 e_inf = 0.045017[2] rms = 0.0167162 e_inf = 0.044046[3] rms = 0.0158089 e_inf = 0.044153[4] rms = 0.0166283 e_inf = 0.044033[5] rms = 0.0152495 e_inf = 0.046928[6] rms = 0.0004666 e_inf = 0.001923[7] rms = 0.0153632 e_inf = 0.043525[8] rms = 0.0142846 e_inf = 0.047231[9] rms = 0.0001769 e_inf = 0.001006

Node adaptive scheme versus non-adaptive with DD:

• Maximum error: 4 × 10−2.

• Graph in a loglog scale, in the x−axis number of nodesand the y−axis time in seconds.

• The node adaptive scheme with domain decompositionmethod requires only an 27% of the time required of thethe non-adaptive scheme.

Conclusions and Further Work

• Based on a quadtree algorithm and a RBF lo-

cal error estimate, a domain decomposition node

adaptive algorithm has been presented.





• The number of subdomaines are dynamically adapted

as the nodes are refined.





• The number of subdomaines are dynamically adapted

as the nodes are refined.

• Numerical experiments on an elliptic and a sta-

tionary convection diffusion problem suggests

that the algorithm can be applied to a large amount

of data.


• As far as to the authors knowledge this is the

first algorithm that combines an adaptive tech-

nique with a domain decomposition algorithm

for scattered data.


• As far as to the authors knowledge this is the

first algorithm that combines an adaptive tech-

nique with a domain decomposition algorithm

for scattered data.

• The method presented in this talk has not

been proved in parallel machines.

• The coordinate bisection algorithm is only a

first attempt to partitioning the domain, more

flexible methods should be considered.

Thank you.

References:

[1] J. Behrens and A. Iske. Grid-free adaptive semi-lagrangian ad-vection using radial basis functions. Computers and Mathematics withApplications, 43(3- 5):319327, 2002.

[2] J. Behrens, A. Iske, and M. Kaser. Adaptive Meshfree Methodof Backward Characteristics for Nonlinear Transport Equations, pages2136. Springer Verlag, 2003.

[3] Phani P. Chinchapatnam, K. Djidjeli, and Prasanth B. Nair. Domaindecomposition for time-dependent problems using radial based mesh-less methods. Numerical Methods for Partial Differential Equations,23(1):3859, 2007.

[4] T.A. Driscoll and A.R.H. Heryudono. Adaptive residual subsam-pling methods for radial basis function interpolation and collocationproblems. Submited to Comp. Math. with Applications.

[5] M.R. Dubal. Domain decomposition and local refinement for mul-tiquadric approximations. I: Second-order equations in one-dimension.. Appl. Sci. Comp., 1(1):146171, 1994.

[6] Y. C. Hon. Multiquadric collocation method with adaptive techniquefor problems with boundary layer. Appl. Sci. Comput., 6(3):17184,1999.

[7] E. J. Kansa and Y. C. Hon. Circumventing the ill-conditioningproblem with multiquadric radial basis functions: Applications to el-liptic partial differential equations. Computers and Mathematics withApplications, 39(78):123 137, 2000.

[8] Jichun Li and C.S. Chen. Some observations on unsymemetricradial basis function collocation methods for convection-diffusion prob-lems. International Journal for Numerical Methods in Engineering,57(8):10851094, 2003.

[9] Jichun Li and Y. C. Hon. Domain decomposition for radial basismeshless methods. Numerical Methods for Partial Differential Equa-tions, 20(3):450 462, 2004.

[10] Leevan Ling and E. J. Kansa. Preconditioning for radial basis func-tions with domain decomposition methods. Mathematical and Com-puter Modelling, 40(13):14131427, 2004.

[11] Leevan Ling and Manfred R. Trummer. Adaptive multiquadriccollocation for boundary layer problem. Journal of Computational andApplied Mathematics, 188(2):265282, 2006.

[12] Jose Antonio Munoz-Gomez, Pedro Gonzalez-Casanova, and Gus-tavo Rodrguez-Gomez. Adaptive node refinement collocation methodfor partial differential equations. In ENC06: Proceedings of the Sev-enth Mexican International Conference on Computer Science, pages7080, Washington, DC, USA, September 2006. IEEE Computer Soci-ety.

[13] Jose Antonio Munoz-Gomez, Pedro Gonzalez-Casanova, and Gus-tavo Rodrguez-Gomez. Domain decomposition by radial basis functions

for time dependent partial differential equations. In ACST06: Proceed-ings of the IASTED International Conference on Advances in ComputerScience and Technology, pages 105109, Anaheim, CA, USA, January2006. ACTA Press.

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[15] A. S. M.Wong, Y. C. Hon, T. S. Li, S. L. Chung, and E. J. Kansa.Multizone decomposition for simulation of time-dependent problemsusing the multiquadric schem. Computers and Mathematics with Ap-plications, 37(8):23 43, 1999.

[16] Zongmin Wu. Dynamically knots setting in meshless method forsolving time dependent propagations equation. Computer Methods inApplied Mechanics and Engineering, 193(12-14):12211229, 2004.

[17] Zongmin Wu. Dynamical knot and shape parameter setting forsimulating shock wave by using multi-quadric quasi-interpolation. En-gineering Analysis with Boundary Elements, 29(4):354358, 2005.

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[20] V. Pereyra , G. Scherer and P. Gonzlez Casanova, Radial FunctionCollocation Solution of Partial Differential Equations in Irregular Do-mains, to appear in: International Journal of Computing Science andMathematics, to appear, (2007).

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