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
Why RBF-PDE methods?
• Classical methods like finite differences, finite
elements, etc. are based on grids.
Introduction
Why RBF-PDE methods?
• Classical methods like finite differences, finite
elements, etc. are based on grids.
• In 2D frequently, 70 % of CPU time of real life
PDE’s problems are used on grid generation.
Introduction
Why RBF-PDE methods?
• Classical methods like finite differences, finite
elements, etc. are based on grids.
• In 2D frequently, 70 % of CPU time of real life
PDE’s problems are used on grid generation.
• In 3D, mesh generation is even more time con-
suming and moving mesh methods are extremely
expensive.
Introduction
Why RBF-PDE methods?
• Classical methods like finite differences, finite
elements, etc. are based on grids.
• In 2D frequently, 70 % of CPU time of real life
PDE’s problems are used on grid generation.
• 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
• No grid is needed to built the RBF interpolant.
• The fact that the RBF depends on the Eu-
clidian norm, implies the Gramm matrix has the
same complexity in 2D and 3D.
Introduction
• No grid is needed to built the RBF interpolant.
• The fact that the RBF depends on the Eu-
clidian norm, implies the Gramm matrix has the
same complexity in 2D and 3D.
• Partial derivatives can be approximated in a
highly accurate way by using RBF.
Introduction
• No grid is needed to built the RBF interpolant.
• The fact that the RBF depends on the Eu-
clidian norm, implies the Gramm matrix has the
same complexity in 2D and 3D.
• Partial derivatives can be approximated in a
highly accurate way by using RBF.
• To solve PDE, several methods have appeared:
Introduction
• No grid is needed to built the RBF interpolant.
• The fact that the RBF depends on the Eu-
clidian norm, implies the Gramm matrix has the
same complexity in 2D and 3D.
• Partial derivatives can be approximated in a
highly accurate way by using RBF.
• 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
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.
2. Node and shape parameter adaptive algorithms,
and
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.
2. Node and shape parameter adaptive algorithms,
and
3. Preconditioning thechniques.
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.
2. Node and shape parameter adaptive algorithms,
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
Up to our knowledge, no algorithm has appeared
which combines Domain decomposition techniques
with node adaptive methods. In this talk:
1. Review Domain decomposition methods.
Outline of the talk
Up to our knowledge, no algorithm has appeared
which combines Domain decomposition techniques
with node adaptive methods. In this talk:
1. Review Domain decomposition methods.
2. Formulate a Node and shape parameter addaptive
algorithms, and
Outline of the talk
Up to our knowledge, no algorithm has appeared
which combines Domain decomposition techniques
with node adaptive methods. In this talk:
1. Review Domain decomposition methods.
2. Formulate a Node and shape parameter addaptive
algorithms, and
3. Formulate a new algorithm which combines these
techniques .
Outline of the talk
Up to our knowledge, no algorithm has appeared
which combines Domain decomposition techniques
with node adaptive methods. In this talk:
1. Review Domain decomposition methods.
2. Formulate a Node and shape parameter addaptive
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:
• Build a numerical scheme based on sets of non
uniform node distributions, rather than on Carte-
sian grids.
• Build a node adaptive thinning RBF method for
PDE’s based on scattered data.
Objective:
• Build a numerical scheme based on sets of non
uniform node distributions, rather than on Carte-
sian grids.
• Build a node adaptive thinning RBF method for
PDE’s based on scattered data.
• Combine a domain decomposition algorithm with
the node adaptive thinning technique to solve the
PDEs problems.
Objective:
• Build a numerical scheme based on sets of non
uniform node distributions, rather than on Carte-
sian grids.
• Build a node adaptive thinning RBF method for
PDE’s based on scattered data.
• 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].
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].
• Evolutionary PDE’s, [15, 3].
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].
• Evolutionary PDE’s, [15, 3].
• In Zhang et. al. [21], a convergence prove is
given for Schwarz additive and multiplicative
methods.
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].
• Evolutionary PDE’s, [15, 3].
• 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.
RBF Domain Decomposition Methods
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.
RBF Domain Decomposition Methods
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.
RBF Domain Decomposition Methods
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.
• 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]:
Schwarz Methods for RBF-PDE
All these methods are based on Schwarz approach,
Zhou et. al. [19]:
• Schwarz additive method: Easer to parallelize
than the multiplicative version.
Schwarz Methods for RBF-PDE
All these methods are based on Schwarz approach,
Zhou et. al. [19]:
• Schwarz additive method: Easer to parallelize
than the multiplicative version.
• Schwarz multiplicative method. Twice faster
than the additive version.
Schwarz Methods for RBF-PDE
All these methods are based on Schwarz approach,
Zhou et. al. [19]:
• Schwarz additive method: Easer to parallelize
than the multiplicative version.
• 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.
Schwarz Methods for RBF-PDE
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.
Schwarz Methods for RBF-PDE
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,
Schwarz Methods for RBF-PDE
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.
Schwarz Methods for RBF-PDE
• 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.
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.
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.
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.
• Construct a local TPS interpolant r2i log(ri)+p1:
Iu(x) based on Nx\x
Adaptive RBF Collocation Method
• 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)
Adaptive RBF Collocation Method
• 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)
• For all x If η(x) > θr refine. If η(x) < θc remove.
Adaptive RBF Collocation Method
• 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)
• 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.
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.
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.
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 2D and for scattered data we find the following
approaches:
RBF Adaptive Methods
• 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.
RBF Adaptive Methods
• 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.
• 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.
RBF Adaptive Methods
• 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.
• 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.
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.
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.
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.
• An initial value of the shape parameter cini is
given, and it is actualized at each iteration by
cnewj = cold
j /2.
Local node refinement and DomainDecomposition Algorithm:
• A recursive coordinate bisecction is use to form
the subdomains.
Local node refinement and DomainDecomposition Algorithm:
• A recursive coordinate bisecction is use to form
the subdomains.
• A given number Nmax is used to determine
whether the subdomain should be divided or not.
Local node refinement and DomainDecomposition Algorithm:
• A recursive coordinate bisecction is use to form
the subdomains.
• A given number Nmax is used to determine
whether the subdomain should be divided or not.
• 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
Local node refinement and DomainDecomposition Algorithm:
• A recursive coordinate bisecction is use to form
the subdomains.
• A given number Nmax is used to determine
whether the subdomain should be divided or not.
• 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.
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
3. While maxη(x) > θr or iterations <50
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
3. While maxη(x) > θr or iterations <50
4. Partition Ω by recursive coordinate bisecction
and form the overlaping zones.
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
3. While maxη(x) > θr or iterations <50
4. Partition Ω by recursive coordinate bisecction
and form the overlaping zones.
5. Solve PDE by multiplicative Schwarz algorithm.
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
3. While maxη(x) > θr or iterations <50
4. Partition Ω by recursive coordinate bisecction
and form the overlaping zones.
5. Solve PDE by multiplicative Schwarz algorithm.
6. Join the solution obtained in each Ωi to built the
soution in Ω.
Local node refinement and DomainDecomposition Algorithm:
1. Define The variables: levelini, cini, θr, θc, Nmax.
2. Built the initial tree and refine two times in the
boundary
3. While maxη(x) > θr or iterations <50
4. Partition Ω by recursive coordinate bisecction
and form the overlaping zones.
5. Solve PDE by multiplicative Schwarz algorithm.
6. Join the solution obtained in each Ωi to built the
soution in Ω.
7. Perform the local refinement of nodes. End While.
Local node refinement and DomainDecomposition Algorithm:
1. The stopping criteria: there are no more nodes
to insert or number of iterations exceeded.
Local node refinement and DomainDecomposition Algorithm:
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.
Local node refinement and DomainDecomposition Algorithm:
1. To join the solution of each sub-domain, a node
is index both to the domain and to the sub-
domain.
Local node refinement and DomainDecomposition Algorithm:
1. To join the solution of each sub-domain, a node
is index both to the domain and to the sub-
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.
Local node refinement and DomainDecomposition Algorithm:
1. To join the solution of each sub-domain, a node
is index both to the domain and to the sub-
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.
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.
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.
• 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.
Conclusions and Further Work
• 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.
Conclusions and Further Work
• 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.
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