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Multiresolution et methodes adaptatives pour les problemesdomines par la convection
Marie PostelLaboratoire Jacques Louis Lions
Mini symposium ”‘Methodes de multiresolution”’Congres SMAI 2009 28 mai 2009
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Outline of the talk
Multiresolution methodsMultiscale analysisAdaptive scheme for hyperbolic PDEs
New developmentsFully adaptive methodsMultiresolution a la Harten
Local Time Stepping Srategies
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Outline
Multiresolution methodsMultiscale analysisAdaptive scheme for hyperbolic PDEs
New developmentsFully adaptive methodsMultiresolution a la Harten
Local Time Stepping Srategies
![Page 4: Multiresolution et methodes adaptatives pour les probl ...Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems January 2223, 2009 Paris, France Promotion](https://reader034.vdocuments.us/reader034/viewer/2022042802/5f40c27f0444360f92094b37/html5/thumbnails/4.jpg)
Adaptive numerical methods for PDEs◮ The solution u is discretized on a non-uniform mesh T in
which the resolution is locally adapted to its singularities(shocks, boundary layers, sharp gradients... ).
◮ Goal: better trade-off between accuracy and CPU/memoryrequirements .
◮ The adaptive mesh is updated based on the a-posterioriinformation gained through the computation
(u0,T0)→ (u1,T1)→ · · · → (un,Tn)
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Adaptive numerical methods for PDEs◮ The solution u is discretized on a non-uniform mesh T in
which the resolution is locally adapted to its singularities(shocks, boundary layers, sharp gradients... ).
◮ Goal: better trade-off between accuracy and CPU/memoryrequirements .
◮ The adaptive mesh is updated based on the a-posterioriinformation gained through the computation
(u0,T0)→ (u1,T1)→ · · · → (un,Tn)
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Adaptive numerical methods for PDEs◮ The solution u is discretized on a non-uniform mesh T in
which the resolution is locally adapted to its singularities(shocks, boundary layers, sharp gradients... ).
◮ Goal: better trade-off between accuracy and CPU/memoryrequirements .
◮ The adaptive mesh is updated based on the a-posterioriinformation gained through the computation
(u0,T0)→ (u1,T1)→ · · · → (un,Tn)
![Page 7: Multiresolution et methodes adaptatives pour les probl ...Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems January 2223, 2009 Paris, France Promotion](https://reader034.vdocuments.us/reader034/viewer/2022042802/5f40c27f0444360f92094b37/html5/thumbnails/7.jpg)
Adaptive numerical methods for PDEs◮ The solution u is discretized on a non-uniform mesh T in
which the resolution is locally adapted to its singularities(shocks, boundary layers, sharp gradients... ).
◮ Goal: better trade-off between accuracy and CPU/memoryrequirements .
◮ The adaptive mesh is updated based on the a-posterioriinformation gained through the computation
(u0,T0)→ (u1,T1)→ · · · → (un,Tn)
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Two typical setups for adaptive methods
Steady state problems
F (u) = 0
the mesh Tn is refined according to local error indicators (forexample based on residual F (un) ) and un → u as n → +∞.
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Two typical setups for adaptive methods
Steady state problems
F (u) = 0
the mesh Tn is refined according to local error indicators (forexample based on residual F (un) ) and un → u as n → +∞.
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Two typical setups for adaptive methods
Steady state problems
F (u) = 0
the mesh Tn is refined according to local error indicators (forexample based on residual F (un) ) and un → u as n → +∞.
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Two typical setups for adaptive methodsSteady state problems
F (u) = 0
the mesh Tn is refined according to local error indicators (forexample based on residual F (un) ) and un → u as n → +∞.
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Two typical setups for adaptive methodsSteady state problems
F (u) = 0
the mesh Tn is refined according to local error indicators (forexample based on residual F (un) ) and un → u as n → +∞.
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Two typical setups for adaptive methods
Evolution problems
∂tu = ǫ(u)
the numerical solution un approximates u(., n∆t) and the meshTn is dynamically updated from time step n to n + 1.
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Two typical setups for adaptive methods
Evolution problems
∂tu = ǫ(u)
the numerical solution un approximates u(., n∆t) and the meshTn is dynamically updated from time step n to n + 1.
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Two typical setups for adaptive methods
Evolution problems
∂tu = ǫ(u)
the numerical solution un approximates u(., n∆t) and the meshTn is dynamically updated from time step n to n + 1.
![Page 16: Multiresolution et methodes adaptatives pour les probl ...Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems January 2223, 2009 Paris, France Promotion](https://reader034.vdocuments.us/reader034/viewer/2022042802/5f40c27f0444360f92094b37/html5/thumbnails/16.jpg)
Two typical setups for adaptive methods
Evolution problems
∂tu = ǫ(u)
the numerical solution un approximates u(., n∆t) and the meshTn is dynamically updated from time step n to n + 1.
![Page 17: Multiresolution et methodes adaptatives pour les probl ...Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems January 2223, 2009 Paris, France Promotion](https://reader034.vdocuments.us/reader034/viewer/2022042802/5f40c27f0444360f92094b37/html5/thumbnails/17.jpg)
Two typical setups for adaptive methods
Evolution problems
∂tu = ǫ(u)
the numerical solution un approximates u(., n∆t) and the meshTn is dynamically updated from time step n to n + 1.
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Multiresolution method
Context: systems of hyperbolic PDEs
∂tu + DivxF (u) = 0
Difficulties:◮ singularities◮ theory : entropy weak solutions◮ numerical analysis : costly schemes and limited order of
convergence
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Multiscale analysis
◮ Answer: adaptive mesh refinement◮ Difficulties:
◮ implementation : displacement of the singularities → mesh◮ convergence analysis
◮ Existing approaches:◮ AMR (Adaptative Mesh Refinement) (Berger, Oliger, ...)◮ Adaptive multiresolution flux evaluation (Harten, Abgrall,
Chiavassa-Donat, ...)◮ Galerkin methods in wavelet spaces
(Bacri-Mallat-Papanicolaou, Dahmen-Cohen-Masson,Maday-Perrier, Bertoluzza, ...)
◮ Fully adaptive multiresolution scheme◮ Harten’s discrete multiresolution framework and link with
wavelet theory◮ Finite volume scheme on a time adaptive grid
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Discrete multiresolution frameworkIn 1D Discretization of a function u(x) by its� point values uj ,k = u(k2−j)
� mean values uj ,k = 2j∫ (k+1)2−j
k2−ju(x)dx
x=0 x=L∆
∆
x
x/2
Dyadic hierarchy of gridsSj , j = 0, . . . , J
In 2D
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Encoding / Decoding (mean value discretization)Projection operator P j
j−1 of Sj on Sj−1: Uj−1 = P jj−1Uj .
j
j−1k
2k2k+1
uj−1,k = 12(uj ,2k + uj ,2k+1)
Prediction operator P j−1j from Sj−1 to Sj : Uj = P j−1
j Uj−1
j−1 j
kk−1 k+1 2k 2k+1
is an approximation of Uj
� local� consistentP j
j−1P j−1j = I
� exact for polynomialsof degree r = 2n
uj ,2k = uj−1,k +
n∑
l=1
γl(uj−1,k+l − uj−1,k−l
)
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Multiscale decomposition
Prediction error:uj ,2k − uj ,2k
(consistence⇒ )= uj ,2k+1 − uj ,2k+1
Details :dj ,k = uj ,2k − uj ,2k
errors on all but onesubdivisions
Details Dj
+Mean values Uj−1
m
Mean values Uj
UJ ⇔ (UJ−1, DJ)
⇔ (UJ−2, DJ−1, DJ)
...
⇔ (U0, D1, . . . , DJ)
= MUJ = (dλ)λ∈∇J
UJ MUJ
Encoding
Decoding
complexity
in O(Card(SJ))
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Biorthogonal wavelet framework
u =∑
λ dλΨλ,
dλ =< u, Ψλ >,Ψλ = 2j/2Ψ(2j .− k),λ = (j , k).
Ψ∼
U2
U1
U0
D2
D1
∣∣dj ,k∣∣ ≤ C2−js
Measure of the local smoothness if u ∈ Cs(Supp Ψj ,k ),Polynomial exactness of degree r with s ≤ r .
Thresholding u → TΛu :=∑
λ∈Λ dλΨλ
Set of significant details Λ = Λε := {λ, |dλ| ≥ ε}.
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0 0.5 1
−0.8
−0.4
0
0.4
x
U
Function U on finest grid
U
0 0.5 10
2
4
6
x
j
significant details
0 0.5 1x
Adaptive grid
0 0.5 1
−0.8
−0.4
0
0.4
x
U
Uε reconstructed on finest grid
UUε
0 0.5 1
−0.8
−0.4
0
0.4
x
U
Uε on adaptive grid
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� Thresholding
Γε = {λ ∈ ∇J , |dλ| ≥ ε|λ|}, εj = 2d(j−J)ε.
Tε(dλ) =
{dλ if λ ∈ Γε
0 otherwise
AεUJ =M−1TεMUJ .
� Adaptive grid Sε = S(Γε)
(dλ)λ∈Γε←→ (Uλ)λ∈Sε
.
Tree structure for the adaptive tree Γε ⇒ Complexity of thecoding / decoding algorithm in O(Card(Γε)).
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Control of error
||AεUJ − UJ ||L1 ≤ Cε
ε−17
10−1
10
−1610
−1210
−810
−410
010
f1f2f3f4
0 100 200 300 400 5000
2
4
6
8x 10−3
smooth function
10 levels
←r=0
←r=2←r=4 ←r=6
Error versus compression
0 50 100 1500
0.002
0.004
0.006
0.008
0.01discontinuous function
10 levels←r=0
←r=2←r=4
←r=6
Error versus compression
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Adaptive scheme for hyperbolic PDEs
∂tu + DivxF (u) = 0.
Reference scheme on finest grid SJ :
Un+1J = EJUn
J
Un+1λ = Un
λ + B(Unν ; ν ∈ Vλ ⊂ SJ), λ ∈ SJ
Harten’s algorithm still on finest grid SJ
Un+1ε,λ = Un
ε,λ + Bε(Unε,ν; ν ∈ Vλ ⊂ SJ), λ ∈ SJ
Fully adaptive algorithm on S(Γnε)
Un+1ε = EεUn
ε .
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Evolution of (Unε , Γn
ε) into (Un+1ε , Γn+1
ε )
� Encoding of thesolution at time tn
� Thresholding to obtainthe tree Γn
ε
� Prediction
� Decoding� Evolution
Solution at time tn
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Ugrid
Details at time tn
-0.4
-0.2
0
0.2
0.4
0 0.2 0.4 0.6 0.8 1
D3D2D1
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Evolution of (Unε , Γn
ε) into (Un+1ε , Γn+1
ε )
� Encoding
� Thresholding to obtainthe tree Γn
ε
� Prediction of the treeΓn+1
ε and grid S(Γn+1ε )
(Harten heuristics ormore refined strategy)
� Decoding� Evolution
Details at time tn
-0.4
-0.2
0
0.2
0.4
0 0.2 0.4 0.6 0.8 1
D3D2D1
Tree Γn+1ε and grid S(Γn+1
ε )
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
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Evolution of (Unε , Γn
ε) into (Un+1ε , Γn+1
ε )
� Encoding
� Thresholding� Prediction
� Decoding on the gridS(Γn+1
ε )� Evolution with fluxescomputed usingreconstructed solution
Tree Γn+1ε and grid S(Γn+1
ε )S
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
Solution at time tn+1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Ugrid
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Error analysis
TheoremCohen, Kaber, Muller, P., Math. Comp., 2003
||UnJ − Un
ε ||L1 ≤ Cnε
Hypotheses◮ More demanding rules for the refinement of the tree◮ Smoothness of the underlying wavelet◮ Scalar equation, on 1D or cartesian multi-D grid◮ Local reconstruction at the finest level
Recent relaxation of the last hypothesis by Hovhannisyan &Muller
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Outline
Multiresolution methodsMultiscale analysisAdaptive scheme for hyperbolic PDEs
New developmentsFully adaptive methodsMultiresolution a la Harten
Local Time Stepping Srategies
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Recent developements and trends
Workshop on Multiresolution and Adaptive Methods forConvection-Dominated ProblemsJanuary 2223, 2009Paris, FrancePromotion of multiresolution and other adaptive techniques forcomplex applications where convection is the prevailingphenomenon.
◮ Robustness of the multiresolution method : wide variety ofextensions
◮ Importance of data structures / parallelisation◮ Problem dependent performances / comparison between
Multiresolution and AMR◮ Local Time Stepping strategies
http://www.ann.jussieu.fr/mamcdp09
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New applications
◮ Incompressible Navier-Stokes equations. Two-dimensional steady
incompressible flow. Muller-Stiriba, Bramkamp-Lamby-Muller
◮ Compressible Navier-Stokes equations. Schneider-Farge-Nguyen,
Chiavassa-Donat-Boiron
◮ Compressible Euler equations. Schneider-Roussel, Muller-Stiriba
◮ Diphasic compressible flows.
Slugging in pipelines. Coquel-Tran-MP-Nguyen-Andrianov
Laser-Induced Cavitation. Bubbles Muller-Bachmann-Kroninger-Kurz-Helluy
◮ Reaction-diffusion equations. 2D thermo-diffusive flames, 3D flame balls.
Schneider-Roussel-Gomes-Domingues
◮ Plasma simulation - Vlasov equation. Sonnendrucker, Campos-Pinto,
Mehrenberger
◮ Shallow water flows Lamby-Muller-Stiriba, Chiavassa-Donat-Gavara
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New tools
◮ Turbulent weakly compressible 3d mixing layer, CoherentVortex Simulation. Schneider & FargePenalisation method around obstacle Chiavassa et al
◮ Anisotropic mesh refinementCohen-Dyn-Hecht-Mirebeau
◮ Treatment of source terms with well balanced schemesChiavassa-Donat-Gavara
◮ Data structures / ParallelisationMuller et al, Sonnendrucker et al
◮ Local Time steppingMuller et al, Schneider et al , Coquel et al, Faille-Nataf
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Data structure
In the fully adaptive scheme, one cell is viewed with respect to◮ its neighbours in the adaptive grid when updating the
solution with the numerical scheme◮ its parents and children when updating the adaptive grid at
each time step
Data structures become crucial with 3D applications and / orwhen going parallel
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Data structure◮ Tree structure (Schneider - Roussel) : Full priority to the
multiresolution◮ Hash tables (Muller - Voss) : compromise between
multiresolution efficiency and implementation of thescheme on the adaptive grid
◮ Sparse data structure (Sonnendrucker - Latu)
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Data structure◮ Tree structure (Schneider - Roussel) : Full priority to the
multiresolution◮ Hash tables (Muller - Voss) : compromise between
multiresolution efficiency and implementation of thescheme on the adaptive grid
◮ Sparse data structure (Sonnendrucker - Latu)
FIG.: Courtesy ¨Mueller et al
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Parallelisation of adaptive methods
Multiple criteria◮ Load balancing partitionning of the adaptive mesh◮ Minimization of the sub domains interfaces◮ Efficiency of the partitionning to be used in a time adaptive
algorithm
Space Filling Curves(Peano-Hilbert)
YODA (Metzmeyer, Hoenen)RAMSES (CEA)
QUADFLOW (Mogosan, Muller)
Graph partitionning(Karypis, Kumar)
GGP, GGGP, BKL, etc.METIS, PETSc
Code Aster (CEA-EDF-Inria)Scotch (Pellegrini)
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Multilevel cost effective techniques
Cost-reduction implementation : NO memory gain! but NOdata-structures needed
◮ Bihari-Harten JCP (1996)SISC (1997)(1D-2D/tensor-product CA-MR) Bihari AIAA (2003)(CA-MR unstructured) FV
◮ Abgrall-Harten SINUM (1996) (CA-MR unstructured) FV◮ Dahmen, Gottschlich-Muller,Muller Num. Math (2000)
(curvilinear meshes, Cell-Average Framework) FV◮ Cohen, Dyn, Kaber, MP JCP (2000) (2D-unstructured) FV◮ Chiavassa-Donat SISC-(2001) (2D/tensor-product PV-MR)
FD
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Recent Applications
FD schemes◮ Well-Balanced (TVDB) schemes for Shallow water flows
(Donat-Gavarra). 2D: CPU Gain from 5.5 to 7 (for gridsfrom 1282 to 5122
◮ Penalization method for compressible Navier Stokes flowsaround obstacle (hig Mach number interactions)[Chiavassa,Donat,Boiron], 2D: CPU Gain from 3 to 5.5 (forgrids from 5122 to 15362)
◮ Propagation of waves in porous medium [Chiavassa,Lombart, Piraux]
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Outline
Multiresolution methodsMultiscale analysisAdaptive scheme for hyperbolic PDEs
New developmentsFully adaptive methodsMultiresolution a la Harten
Local Time Stepping Srategies
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From multiresolution to locally adaptive time stepping
Standard MR scheme◮ At a given time, same time step used for all cells
◮ Time step ruled by CFL condition ∆tn < minj
∆xj
µ(Unj )
.
Principle of the LTS strategy: use a time-step adapted to thelocal size of the cell constant λ = ∆t/∆x(Berger, Colella ’89, Mueller, Stiriba ’06)
∆2∆
2∆∆t
t
x x
Goal : further reduction of # calls to flux functions andexpensive state laws⇒ CPU ↓
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Evolution of the solution at intermediate time
Uni
Un+1i
Un+2i
∆tn+1
∆tn
∆tn+1,1
∆tn,4
∆tn,3
∆tn,3
∆tn,1
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Synchronization at intermediate times
Osher-Sanders strategy
x
t
Unk ,j−2 U
nk ,j−1
Un+1k ,j−1 U
n+1k+1,2j
Un+1k+1,2j+1
Unk+1,2j
Unk+1,2j+1
Un+1/2k+1,2j+1
Un+1/2k+1,2j
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Synchronization at intermediate times
Schneider-Roussel-Gomes-Domingues strategy
x
t
Unk ,j−2
Unk ,j−1
Un+1k ,j−1U
n+1k+1,2j
Unk+1,2j
Unk+1,2j+1 U
n+1/2k+1,2j+1
Un+1/2k+1,2j
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Synchronization at intermediate times
Muller & Stiriba strategy
x
t
Unk ,j−2
Un+1k ,j−2
Unk ,j−1 U
nk+1,2j
Un+1/2k ,j−1 U
n+1/2k+1,2j
Un+1k+1,2jU
n+1k ,j−1
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Local Time Stepping (LTS)Computation of the time-step
DefinitionsMacro time step ∆tn ↔ largest cellsMicro time step ∆tn,i
K ↔ smallest cells∆tn =
2K∑
i=1
∆tn,iK
Unj
Un+1j
Un+2j
∆tn+1
∆tn
∆tn+1,1K
∆tn,4K
∆tn,3K
∆tn,2K
∆tn,1K
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Local Time Stepping (LTS)Computation of the time-step
◮ The micro time-steps decrease during each macrotime-step
◮ They are updated along with the solution while ensuringthe stability condition
Unj
Un+1j
Un+2j
∆tn+1
∆tn
∆tn+1,1K
∆tn,4K
∆tn,3K
∆tn,2K
∆tn,1K
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Local Time Stepping (LTS)Computation of the time-step
◮ The micro time-steps decrease during each macrotime-step
◮ They are updated along with the solution while ensuringthe stability condition
◮ The first micro time-step ∆tn,1 < minj
∆xj
µ(Unj )
.
◮ The others
∆tn,p < min
(∆tn,p−1
K , minj
∆xj
µ(Un,p−1j )
)
where Un,p−1j is the updated solution at time tn +
∑p−1i=1 ∆tn,i
K
◮ The macro time-step ∆tn =
2K∑
i=1
∆tn,i
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Test case with smoothly varying initial condition
400
450
500
550
0 10000 20000 30000 0
2
4
6
ρ
x
UNIMRLTSt=0
LTS grid
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Error versus computing time gain - simplistic state law
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Error versus number of calls to state laws gain
1e-07
1e-06
1e-05
1e-04
0.001
0.01
0.1 1 10 100 1000
E(Y)
# State laws gain
MRLTS
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Conclusions and perspectives
◮ Very active and uprising field◮ Need for strong collaboration between computer scientists
and mathematicians◮ Comparison between AMR and MR◮ Visit the website
http://www.ann.jussieu.fr/mamcdp09