Optimized Domain Desomposition Methodes with HPDDM Libraryhecht/ftp/ff++days/2016/pdf/RH.pdf ·...

Transcript of Optimized Domain Desomposition Methodes with HPDDM Libraryhecht/ftp/ff++days/2016/pdf/RH.pdf ·...

Optimized Domain Desomposition Methodes withHPDDM Library

Ryadh Haferssas 1 Pierre Jolivet 2 Frédéric Nataf 1

2IRIT-CNRS

1LJLL-CNRS-INRIA(Alpines)

December 9, 2016
Intro to DDM & improvements Fields of Application & Numerical Results Recycling Krylov and Application to Navier-Stokes Conclusion

1 Introduction to DDM & improvements

2 Fields of Application & Numerical Results

3 Recycling Krylov and Application to Navier-Stokes

4 Conclusion

R.H ODDM in HPDDM 2/ 31

Outline




4 Conclusion


Why Domain Decomposition Methods ?

Find u P Vh such that : @v P VhaΩpu, vq “ `pvq

ùñ Au “ F P Rn.

Darcy aΩpu, vq “ż

Ω

κ∇u ¨ ∇ v dx

Elasticty aΩpu, vq “ż

Ω

C : εpuq : εpvq dx

Purpose

Solve AU “ F . Achieve robustness with regard to the irregularity of thecoefficient distribution

Achieve a strong and weak scalability when solving with thousands ofprocessors

Tools

A good mathematical foundation theory & HPDDM Library: a parallelframework




ùñ Au “ F P Rn.


Ω

κ∇u ¨ ∇ v dx


Ω


Purpose



Tools





ùñ Au “ F P Rn.


Ω

κ∇u ¨ ∇ v dx


Ω


Purpose



Tools





ùñ Au “ F P Rn.


Ω

κ∇u ¨ ∇ v dx


Ω


Purpose



Tools





ùñ Au “ F P Rn.


Ω

κ∇u ¨ ∇ v dx


Ω


Purpose



Tools




How can we solve a large sparse system Au “ F P Rn ?

Memory consumptionRobustnessParallelizable

Direct Methods

Iterative Methods Memory consumptionRobustnessParallelizable

DDM “Hybrid” methods



How can we solve a large sparse system Au “ F P Rn ?

Memory consumptionRobustnessParallelizable

Direct Methods

Iterative Methods Memory consumptionRobustnessParallelizable

DDM “Hybrid” methods


Original Schwarz DDM (H.A. Schwarz, 1870)

#

´∆u “ 0 in Ωu “ g on BΩ

ùñ Au “ F P Rn.⌦1⌦2

Given pun1 , un2 q, find pun`11 , un`12 q such thatˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

´∆un`11 “ 0 in Ω1un`11 “ g on BΩ1 X BΩun`11 “ un2 on ΩzΩ1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

´∆un`12 “ 0 in Ω2un`12 “ g on BΩ2 X BΩun`12 “ un`11 on ΩzΩ2

[Martin J. Gander and Wanner 2014]


Original Schwarz DDM (H.A. Schwarz, 1870)

#

´∆u “ 0 in Ωu “ g on BΩ

ùñ Au “ F P Rn.⌦1⌦2

Given pun1 , un2 q, find pun`11 , un`12 q such thatˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

´∆un`11 “ 0 in Ω1un`11 “ g on BΩ1 X BΩun`11 “ un2 on ΩzΩ1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

´∆un`12 “ 0 in Ω2un`12 “ g on BΩ2 X BΩun`12 “ un`11 on ΩzΩ2

[Martin J. Gander and Wanner 2014]


Schwarz DDM Notations & definitions

From finite elements spaces :

Global Vh “ span

φk : k “ 1, . . . , n(

ÑN

Local VhpΩiq “ span!

φk|Ωi : mespsupppφkqq X Ωi ‰ 0)

ÑNi

N “Nď

i“1Ni


Schwarz DDM Notations & definitions

From finite elements spaces :

Global Vh “ span

φk : k “ 1, . . . , n(

ÑN

Local VhpΩiq “ span!

φk|Ωi : mespsupppφkqq X Ωi ‰ 0)

ÑNi

N “Nď

i“1Ni


Interpolation Operators :

Global to Local Ri : R#N ÝÑ R#Ni (Restriction)Local to Global RTi : R

ᵀi : R#Ni ÝÑ R#N (Prolongation)

Duplicated unknowns are coupled via apartition of unity:

Di : R#Ni ÝÑ R#NiNÿ

i“1Rᵀi Di Ri “ Id.

Ω1 Ω21

2

1

2

Then

un`1 “Nÿ

i“1Rᵀi Diu

n`1i


Interpolation Operators :

Global to Local Ri : R#N ÝÑ R#Ni (Restriction)Local to Global RTi : R

ᵀi : R#Ni ÝÑ R#N (Prolongation)

Duplicated unknowns are coupled via apartition of unity:

Di : R#Ni ÝÑ R#NiNÿ

i“1Rᵀi Di Ri “ Id.

Ω1 Ω21

2

1

2

Then

un`1 “Nÿ

i“1Rᵀi Diu

n`1i


To solve Au “ F Schwarz methods can be viewed as preconditioners for a fixedpoint algorithm:

un`1 “ un ` M´1pf ´ Aunq.

Where the Schwarz preconditioner can be

Classical :

M´1ASM,1 :“Nř

i“1Rᵀi Ai

´1Ri with Ai “ RiARᵀi [Toselli and Widlund 2005]

M´1RAS,1 :“Nř

i“1Rᵀi DiA

´1i Ri [Cai and Sarkis 1999]


To solve Au “ F Schwarz methods can be viewed as preconditioners for a fixedpoint algorithm:

un`1 “ un ` M´1pf ´ Aunq.

Where the Schwarz preconditioner can be

Classical :

M´1ASM,1 :“Nř

i“1Rᵀi Ai

´1Ri with Ai “ RiARᵀi [Toselli and Widlund 2005]

M´1RAS,1 :“Nř

i“1Rᵀi DiA

´1i Ri [Cai and Sarkis 1999]


P-L. Lions algorithm

´∆un`11 “ f in Ω1un`11 “ 0 on BΩ1 X BΩ

´

BBn1 ` α

¯

un`11 “´

BBn1 ` α

¯

un2 on BΩ1 X Ω2 ,

and´∆un`11 “ f in Ω2

un`11 “ 0 on BΩ2 X BΩ´

BBn2 ` α

¯

un`12 “´

BBn2 ` α

¯

un1 on BΩ2 X Ω1

Optimized :

M´1ORAS,1 :“Nř

i“1Rᵀi DiBi

´1Ri [St-Cyr, M. J. Gander, and Thomas 2007]

M´1SORAS,1 :“Nř

i“1Rᵀi DiBi

´1DiRi [Haferssas, Jolivet, and Nataf 2015]

R.H ODDM in HPDDM 10 / 31



´

BBn1 ` α

¯

un`11 “´

BBn1 ` α

¯




BBn2 ` α

¯

un`12 “´

BBn2 ` α

¯

un1 on BΩ2 X Ω1

Optimized :

M´1ORAS,1 :“Nř

i“1Rᵀi DiBi


M´1SORAS,1 :“Nř

i“1Rᵀi DiBi





´

BBn1 ` α

¯

un`11 “´

BBn1 ` α

¯




BBn2 ` α

¯

un`12 “´

BBn2 ` α

¯

un1 on BΩ2 X Ω1

Optimized :

M´1ORAS,1 :“Nř

i“1Rᵀi DiBi


M´1SORAS,1 :“Nř

i“1Rᵀi DiBi



One-Level methods are not scalable

Figure: 2D domain partitioned with METIS

AS RAS ORAS SORAS# DOFs # subdom # iterations # iterations # iterations # iterations

66703 16 140 140 96 57130433 32 245 216 146 109266812 64 383 312 245 203541838 128 566 460 480 398

Table: 2D Elasticity: number of GMRES iterations for compressible case with E “ 107and ν “ 0.3


One-Level methods are not scalable

Figure: 2D domain partitioned with METIS

AS RAS ORAS SORAS# DOFs # subdom # iterations # iterations # iterations # iterations

66703 16 140 140 96 57130433 32 245 216 146 109266812 64 383 312 245 203541838 128 566 460 480 398

Table: 2D Elasticity: number of GMRES iterations for compressible case with E “ 107and ν “ 0.3


Two Level Domain Decomposition Methods

Given

VH :“ spanpZq

ó

RH “ Zᵀ : R#N ÝÑ R#Z

E :“ RHARᵀHlooooooomooooooon

coarse problem, E much smaller than A

Enrich the one level preconditioner with Z (ie Q “ RᵀHE´1RH)

:

M´12,AD :“ Q ` M´1

M´12,A´DEF1 :“ M´1pI ´ AQq ` Q

M´12,A´DEF2 :“ pI ´ QAqM´1 ` Q [Tang, Nabben, Vuik, and Erlangga 2009]

M´12,BNN :“ pI ´ QAqM´1pI ´ AQq ` Q [Mandel 1992]

what does the space VH contain?



Given

VH :“ spanpZq

ó




Enrich the one level preconditioner with Z (ie Q “ RᵀHE´1RH) :

M´12,AD :“ Q ` M´1







Given

VH :“ spanpZq

ó





M´12,AD :“ Q ` M´1







Given

VH :“ spanpZq

ó





M´12,AD :“ Q ` M´1







Given

VH :“ spanpZq

ó





M´12,AD :“ Q ` M´1







Given

VH :“ spanpZq

ó





M´12,AD :“ Q ` M´1






GenEO I approach to build VH

Find the eigenpairs pλi,k, Ui,kq

ANi Ui,k “ λi,kDiAiDiUi,k

ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

With :

ANi local matrix, resulting from the local bilinear form,with Neumann boundary condition on the interfaces.

Ri,0 : Ωi ÝÑ Ω˝i “Ť

j‰ipΩi

Ş

Ωjq a restriction operator.

Choose a number of eigenmodes τi for each subdomain then define

Wi “ rDiUi,1, DiUi,2, . . . , DiUi,τis

FinallyZ “ rW1,W2, . . . ,WN s





ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

With :



j‰ipΩi

Ş









ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

With :



j‰ipΩi

Ş






Theoretical result for GenEO I

Theorem (Spillane, Dolean, Hauret, Nataf, Pechstein, Scheichl)

If for all j: 0 ă λj,mj`1 ă 8:

κpM´1AS,2Aq ď p1 ` k0q”

2 ` k0 p2k0 ` 1q´

1 ` τ¯ı

where :

k0 the maximum multiplicity of the interaction between subdomains.

Parameter τ can be chosen arbitrarily small at the expense of a largecoarse space.

[Spillane et al. 2014]


ORAS: Optimized RAS

Optimized RAS :

P.L. Lions algorithm at the continuous level (partial differential equation)

Algebraic formulation for overlapping subdomainsñ Let Bi be the matrix of the Robin subproblem in each subdomain 1 ď i ď N ,define M´1ORAS :“

řNi“1 R

Ti DiB

´1i Ri ,

[St-Cyr, M. J. Gander, and Thomas 2007]

Symmetric variant M´1SORAS,1 :“Nř

i“1Rᵀi DiBi

´1DiRi

Adaptive Coarse space with prescribed targeted convergence rateñ ???


ORAS: Optimized RAS

Optimized RAS :


Algebraic formulation for overlapping subdomains

ñ Let Bi be the matrix of the Robin subproblem in each subdomain 1 ď i ď N ,define M´1ORAS :“

řNi“1 R

Ti DiB

´1i Ri ,



i“1Rᵀi DiBi

´1DiRi



ORAS: Optimized RAS

Optimized RAS :



řNi“1 R

Ti DiB

´1i Ri ,



i“1Rᵀi DiBi

´1DiRi



ORAS: Optimized RAS

Optimized RAS :



řNi“1 R

Ti DiB

´1i Ri ,



i“1Rᵀi DiBi

´1DiRi



ORAS: Optimized RAS

Optimized RAS :



řNi“1 R

Ti DiB

´1i Ri ,



i“1Rᵀi DiBi

´1DiRi

Adaptive Coarse space with prescribed targeted convergence rateñ

???


ORAS: Optimized RAS

Optimized RAS :



řNi“1 R

Ti DiB

´1i Ri ,



i“1Rᵀi DiBi

´1DiRi



SORAS-GenEO II approach to build VHFind the eigenpairs pλi,k, Ui,kq

ANi Ui,k “ λi,kBiUi,k

Find the eigenpairs pµi,k, Vi,kq

BiVi,k “ µi,kDiAiDiVi,k

ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :

Bi the local matrix, resulting from the local bilinear form withOptimized interface conditions

Choose τi and γi for each subdomain then define

Wi “ rDiUi,1 DiUi,2 . . . DiUi,τis And Hi “ rDiVi,1 DiVi,2 . . . DiVi,γis

Zτ “ rW1 W2 . . . WN s And Zγ “ rH1 H2 . . . HN s

Z “ rZτ Zγs






ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :





Z “ rZτ Zγs






ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :



Wi “ rDiUi,1 DiUi,2 . . . DiUi,τis

And Hi “ rDiVi,1 DiVi,2 . . . DiVi,γis


Z “ rZτ Zγs






ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :





Z “ rZτ Zγs






ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :





Z “ rZτ Zγs






ˇ

ˇ

ˇ

ˇ

ˇ

Solved by ARPACK

(Concurrently)

where :





Z “ rZτ ZγsR.H ODDM in HPDDM 16 / 31

Theoretical results for SORAS-GenEO II

Theorem (H., Jolivet and Nataf, 2015)

Let γ and τ be user-defined targets. Then, the eigenvalues of the two-levelSORAS-GenEO II preconditioned system satisfy the following estimate

1

1 ` k1τď κpM´1SORAS,2 Aq ď maxp1, k0 γq

where :

k0 the maximum number of neighbors.

k1 the maximum multiplicity.

τ and γ a user-defined thresholds.


Outline




4 Conclusion


Fields of Application

Compressible elasticity equationż

Ω

2µ εpuq : εpvqdx `ż

Ω

λ∇ ¨ puq∇ ¨ pvqdx “ż

Ω

fdx vdx `ż

ΓNg ¨ v

Nearly-incompressible elasticity equations$

’

’

’

’

&

’

’

’

’

%

2

ż

Ω

µεpuq : εpvqdx ´ż

Ω

p∇ ¨ pvqdx “ż

Ω

fvdx `ż

ΓNg ¨ v,

´ż

Ω

∇ ¨ puqqdx ´ż

Ω

1

λpqdx “ 0

σS

puq.n ` Lpαqu “ 0. on BΩizBΩWhere L is constructed from the Lamé coefficient of the material and it is definedas follows

Lpα, λ, µq :“ 2αµp2µ ` λqλ ` 3µ


Fields of Application

Compressible elasticity equationż

Ω

2µ εpuq : εpvqdx `ż

Ω

λ∇ ¨ puq∇ ¨ pvqdx “ż

Ω

fdx vdx `ż

ΓNg ¨ v

Nearly-incompressible elasticity equations$

’

’

’

’

&

’

’

’

’

%

2

ż

Ω

µεpuq : εpvqdx ´ż

Ω

p∇ ¨ pvqdx “ż

Ω

fvdx `ż

ΓNg ¨ v,

´ż

Ω

∇ ¨ puqqdx ´ż

Ω

1

λpqdx “ 0

σS

puq.n ` Lpαqu “ 0. on BΩizBΩWhere L is constructed from the Lamé coefficient of the material and it is definedas follows

Lpα, λ, µq :“ 2αµp2µ ` λqλ ` 3µ


Stokes equations

$

’

’

’

’

&

’

’

’

’

%

ż

Ω

2µεpuq : εpvqdx ´ż

Ω

p∇ ¨ pvqdx “ż

Ω

fvdx,

ż

Ω

∇ ¨ puqqdx “ 0

σF

puq.n ` Lpαqu “ 0. on BΩizBΩ

Where L is constructed from viscosity coefficient of the fluid and it is defined asfollows

Lpα, λ, µq :“ 2αµp2µ ` λqλ ` 3µ with λ “ 10

8

Diffusion equationż

Ω

κ∇u ¨ ∇ v dx “ż

Ω

f vdx


Stokes equations

$

’

’

’

’

&

’

’

’

’

%

ż

Ω

2µεpuq : εpvqdx ´ż

Ω

p∇ ¨ pvqdx “ż

Ω

fvdx,

ż

Ω

∇ ¨ puqqdx “ 0

σF

puq.n ` Lpαqu “ 0. on BΩizBΩ

Where L is constructed from viscosity coefficient of the fluid and it is defined asfollows

Lpα, λ, µq :“ 2αµp2µ ` λqλ ` 3µ with λ “ 10

8

Diffusion equationż

Ω

κ∇u ¨ ∇ v dx “ż

Ω

f vdx


HPDDM Library - P. Jolivet & F. Nataf

An implementation of several Domain Decomposition Methods :

One-and two-level Schwarz methods

The Finite Element Tearing and Interconnecting (FETI) method

Balancing Domain Decomposition (BDD) method

Library written in C++11 MPI and :

Linked with automatic partitioners (METIS & SCOTCH).

Linked with BLAS & LAPACK.

Linked with direct solvers (MUMPS, SuiteSparse, MKL PARDISO, PASTIX).

Linked with eigenvalue solver (ARPACK).

Interfaced with discretisation kernel FreeFem++ & FEEL++


Machine used for scaling tests

Curie

5,040 compute nodes

2 eight-core Intel Sandy [email protected] GHz per node

1.7 PFLOPs peak performance

Turing, IDRIS-Genci project

IBM Blue Gene/Q

6144 compute nodes (16 core per node @1.6 GHZ )

1.258 PFLOPs peak performance


Weak scalability (Nearly-Incompressible linear elasticity)

”Sandwich” problem discretized by P3zP2(2D), P2zP1(3D)Automatic mesh partition with METIS, 1 subdomain/MPI process, 2 OpenMPthreads/MPI process

256 512 1 0242 048

4 0968 192

0%

20%

40%

60%

80%

100%

# of processes

Effici

ency

rela

tive

to25

6pr

oces

ses

3D2D

22

704

#of

d.o.

f.—in

mill

ions

6

197


Strong scalability test (Stokes problem)

Led Driven cavity discretized by P2zP1 FE 50M d.o.f. (3D), 100M d.o.f. (2D)Automatic mesh partition with METIS

1 0242 048

4 0968 192

40

100

200

500

# of processes

Run

time

(sec

onds

)Linear speedup3D 2D


Outline




4 Conclusion


Incompressible Navier-Stokes in Turek benchmark

$

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

%

ρBuBt ` ρ u ¨ ∇u ´ ∇ ¨ σF pu, pq “ f in Ω ˆ p0, T q

∇ ¨ u “ 0 in Ω ˆ p0, T qu “ g on ΓD ˆ p0, T q

σF

pu, pqn “ h on ΓN ˆ p0, T qup0q “ u0 in Ω ˆ 0

with BDF2$

’

&

’

%

ρ 3un`1´4un`1`un`1

2dt ` ρp2un ´ un´1q ¨ ∇un`1 ´ µ∆un`1 ` ∇pn`1 “ f in Ω

∇ ¨ un`1 “ 0 in ΩBoundary conditions

A “„

H LT

L 0

„

UP

n`1“ F,

with H “ 1∆tM ` A ` CpUnq


DDM preconditionning & Krylov recyling

AnUn “ Fn n “ 1, 2, ...

M´1SORAS,1 :“Nÿ

i“1RTi DiB

´1i DiRi

M´1ORAS,1 :“Nÿ

i“1RTi DiB

´1i Ri

Use GCRO-DR in [Parks et al. 2006] asimplemented in [Jolivet and Tournier2016]

0 20 40 60 80 10010−810−710−610−510−410−410−310−210−1

# of iterations

Relativ

eresidu

alerror

GMRES-(t0)GCRODR-t1GCRODR(t2)

Figure: GMRES iterations - Navier-Stokesproblem with 2 millions d.o.f solved on 128proc for 3 time steps.


Weak scalability - Incompressible Navier-Stokes in Turekbenchmark

256 512 1 0242 048

4 0920%

20%

40%

60%

80%

100%

# of processes

Efficiency

relativ

eto

256processes

3D 13

205

#of

d.o.f.—

inmillions

13

205

CD “ 0.31 P r0.29, 0.31s CL “ ´0.01 P r´0.01,´0.008s Strouhal “ 0.36


Strong scalability - Incompressible Navier-Stokes

Turek benchmark discretized by P2zP1 FE, 87M d.o.f. (3D)automatic mesh partition with METIS

1 0242 048

4 0968 192

16 382

10

20

30

60

120

# of processes

Runtim

e(secon

ds)

Linear speedup3D


Outline




4 Conclusion


Conclusion

Summary

SORAS preconditioner

M´1SORAS :“Nÿ

i“1RTi DiB

´1i DiRi

is amenable to a fruitful theory for OSM

Using two generalized eigenvalue problems, we are able to achieve a targetedconvergence rate for OSM

Nonlinear time dependent problem

Freely available via HPDDM library or FreeFem++

Future work

Another look at parameter α optimization

Multilevel extension of the coarse operator

Recycle coarse spaceR.H ODDM in HPDDM 31 / 31
References

Cai, Xiao-Chuan and Marcus Sarkis (1999). “A restricted additive Schwarzpreconditioner for general sparse linear systems”. In: SIAM Journal onScientific Computing 21, pp. 239–247.

St-Cyr, A., M. J. Gander, and S. J. Thomas (2007). “Optimizedmultiplicative, additive, and restricted additive Schwarz preconditioning”. In:SIAM J. Sci. Comput. 29.6, 2402–2425 (electronic). issn: 1064-8275. doi:10.1137/060652610. url: http://dx.doi.org/10.1137/060652610.

Gander, Martin J. and Gerhard Wanner (2014). “The Origins of theAlternating Schwarz Method”. In: Domain Decomposition Methods inScience and Engineering XXI. Springer, pp. 487–495.

Haferssas, Ryadh, Pierre Jolivet, and Frdric Nataf (2015). “A robust coarsespace for optimized Schwarz methods: SORAS-GenEO-2”. In: C. R. Math.Acad. Sci. Paris 353.10, pp. 959–963. issn: 1631-073X. doi:10.1016/j.crma.2015.07.014. url:http://dx.doi.org/10.1016/j.crma.2015.07.014.


http://dx.doi.org/10.1137/060652610http://dx.doi.org/10.1137/060652610http://dx.doi.org/10.1016/j.crma.2015.07.014http://dx.doi.org/10.1016/j.crma.2015.07.014
References

Jolivet, Pierre and Pierre-Henri Tournier (2016). “Block Iterative Methodsand Recycling for Improved Scalability of Linear Solvers”. In: Proceedings ofthe 2016 International Conference for High Performance Computing,Networking, Storage and Analysis. SC16. IEEE.

Mandel, Jan (1992). “Balancing domain decomposition”. In: Comm. onApplied Numerical Methods 9, pp. 233–241.

Parks, Michael L., Eric de Sturler, Greg Mackey, Duane D. Johnson, andSpandan Maiti (2006). “Recycling Krylov subspaces for sequences of linearsystems”. In: SIAM J. Sci. Comput. 28.5, 1651–1674 (electronic). issn:1064-8275. doi: 10.1137/040607277. url:http://dx.doi.org/10.1137/040607277.

Spillane, N., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl(2014). “Abstract robust coarse spaces for systems of PDEs via generalizedeigenproblems in the overlaps”. In: Numer. Math. 126.4, pp. 741–770. issn:0029-599X. doi: 10.1007/s00211-013-0576-y. url:http://dx.doi.org/10.1007/s00211-013-0576-y.

Tang, J.M., R. Nabben, C. Vuik, and Y.A. Erlangga (2009). “Comparison oftwo-level preconditioners derived from deflation, domain decomposition andmultigrid methods”. In: Journal of Scientific Computing 39.3, pp. 340–370.


http://dx.doi.org/10.1137/040607277http://dx.doi.org/10.1137/040607277http://dx.doi.org/10.1007/s00211-013-0576-yhttp://dx.doi.org/10.1007/s00211-013-0576-y
References

Toselli, Andrea and Olof Widlund (2005). Domain Decomposition Methods -Algorithms and Theory. Vol. 34. Springer Series in ComputationalMathematics. Springer.

References

Weak scaling (Nearly-Incompressible linear elasticity)-time

N Factorization Deflation Solution # of it. Total # of d.o.f.

3D

1 024 79.2 s 229.0 s 76.3 s 45 387.5 s

50.63 · 1062 048 29.5 s 76.5 s 34.8 s 42 143.9 s4 096 11.1 s 45.8 s 19.8 s 42 80.9 s8 192 4.7 s 26.1 s 14.9 s 41 56.8 s

2D

1 024 5.2 s 37.9 s 51.5 s 51 95.6 s

100.13 · 1062 048 2.4 s 19.3 s 22.1 s 42 44.5 s4 096 1.1 s 10.4 s 10.2 s 35 22.6 s8 192 0.5 s 4.6 s 6.9 s 38 12.7 s

References

Weak scaling (Nearly-Incompressible linear elasticity), -time

1 subdomain/MPI process, 2 OpenMP threads/MPI process.

N Factorization Deflation Solution # of it. Total # of d.o.f.

3D

256 25.2 s 76.0 s 37.2 s 46 145.2 s 6.1 · 106512 26.5 s 81.1 s 39.8 s 47 155.1 s 12.4 · 106

1 024 29.2 s 82.6 s 41.7 s 45 165.5 s 25.0 · 1062 048 26.9 s 83.5 s 46.3 s 47 171.0 s 48.8 · 1064 096 28.3 s 88.8 s 54.5 s 53 177.7 s 97.9 · 1068 192 29.0 s 78.3 s 79.8 s 60 196.1 s 197.6 · 106

2D

256 4.8 s 72.9 s 39.9 s 46 123.9 s 22.1 · 106512 4.7 s 65.9 s 45.0 s 51 121.3 s 44.0 · 106

1 024 4.8 s 70.0 s 46.1 s 51 127.0 s 88.3 · 1062 048 4.8 s 69.0 s 46.5 s 51 127.4 s 176.8 · 1064 096 4.8 s 65.8 s 52.8 s 56 132.6 s 351.0 · 1068 192 4.8 s 65.4 s 53.0 s 54 134.8 s 704.1 · 106

References

Weak scaling (Incompressible Navier-Stokes in Turekbenchmark, with FreeFem++ and HPDDM)

N Factorization GCRODR # of it. Total # of d.o.f.

3D256 28.2 s 17.0 s 29 67.9 s 13.7 · 106512 30.5 s 16.7 s 28 68.1 s 26.0 · 1061 024 30.6 s 19.7 s 35 70.8 s 50.9 · 1062 048 30.9 s 24.0 s 41 75.7 s 103.6 · 1064 092 28.6 s 28.5 s 57 79.8 s 205.8 · 106

References

Strong scaling (Incompressible Navier-Stokes in Turekbenchmark, with FreeFem++ and HPDDM)

N Factorization GCRODR # of it. Total Step # of d.o.f.

3D

1 024 70.4 s 27.2 s 36 129.7 s

87.11 · 1062 048 23.7 s 16.7 s 46 59.0 s4 096 8.1 s 10.3 s 57 30.3 s8 192 3.6 s 7.0 s 66 20.4 s

3D

16 384 1.5 s 4.7 s 75 13.1 s

87.11 · 106


Introduction to DDM & improvementsFields of Application & Numerical ResultsRecycling Krylov and Application to Navier-StokesConclusion

Optimized Domain Desomposition Methodes with HPDDM Libraryhecht/ftp/ff++days/2016/pdf/RH.pdf ·...

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Transcript of Optimized Domain Desomposition Methodes with HPDDM Libraryhecht/ftp/ff++days/2016/pdf/RH.pdf ·...