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Multilevel domain decomposition at extreme scales S. Badia, A. - - PowerPoint PPT Presentation

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Multilevel domain decomposition at extreme scales S. Badia, A. Martin, J. Principe Universitat Politcnica de Catalunya & CIMNE Jeju, July 7th, 2015 0 / 24 Outline 1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4

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SLIDE 1

Multilevel domain decomposition at extreme scales

  • S. Badia, A. Martin, J. Principe

Universitat Politècnica de Catalunya & CIMNE

Jeju, July 7th, 2015

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slide-2
SLIDE 2

Outline

1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4 Conclusions

0 / 24

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SLIDE 3

Outline

1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4 Conclusions

0 / 24

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SLIDE 4

Current trends of supercomputing

  • Transition from today’s 10 Petaflop/s supercomputers (SCs)
  • ... to exascale systems w/ 1 Exaflop/s expected in 2020
  • × 100 performance based on concurrency (not higher freq)
  • Future: Multi-Million-core (in broad sense) SCs

1 / 24

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SLIDE 5

Current trends of supercomputing

  • Transition from today’s 10 Petaflop/s supercomputers (SCs)
  • ... to exascale systems w/ 1 Exaflop/s expected in 2020
  • × 100 performance based on concurrency (not higher freq)
  • Future: Multi-Million-core (in broad sense) SCs

1 / 24

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SLIDE 6

Weakly scalable solvers

  • This talk: One challenge, weakly

scalable algorithms

Weak scalability

If we increase X times the number of processors, we can solve an X times larger problem

  • Key property to face more

complex problems / increase accuracy

Source: Dey et al, 2010 Source: parFE project 2 / 24

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SLIDE 7

Scalable linear solvers (AMG)

  • Most scalable solvers for CSE are parallel AMG (Trilinos [Lin, Shadid,

Tuminaro, ...], Hypre [Falgout, Yang,...],...)

  • Hard to scale up to largest SCs today (one million cores, < 10 PFs)
  • Problems: large communication/computation ratios at coarser levels,

densification coarser problems,...

3 / 24

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SLIDE 8

Multilevel framework

  • Propose a highly scalable implementation of Multilevel DD methods

(MLBDDC [Mandel et al’08])

  • MLDD based on a hierarchy of meshes/functional spaces
  • It involves local subdomain problems at all levels (L1, L2, ...)

FE mesh Subdomains (L1) Subdomains (L2)

4 / 24

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SLIDE 9

Outline

1 Motivation I: Develop a multilevel framework suitable for extremely

scalable implementations

2 Motivation II: Apply the multilevel framework for scalable linear algebra

(MLBDDC)

5 / 24

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SLIDE 10

Outline

1 Motivation I: Develop a multilevel framework suitable for extremely

scalable implementations

2 Motivation II: Apply the multilevel framework for scalable linear algebra

(MLBDDC)

5 / 24

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SLIDE 11

Outline

1 Motivation I: Develop a multilevel framework suitable for extremely

scalable implementations

2 Motivation II: Apply the multilevel framework for scalable linear algebra

(MLBDDC)

All implementations in FEMPAR (in-house code) to be dis- tributed as open-source SW soon*

* Funded by Proof of Concept Grant 640957 - FEXFEM: On a free open source extreme scale finite element software

5 / 24

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SLIDE 12

Outline

1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4 Conclusions

5 / 24

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SLIDE 13

Premilinaries

  • Element-based (non-overlapping DD) distribution (+ limited ghost info)

Th T 1

h , T 2 h , T 3 h

˜ T 1

h

  • Gluing info based on objects
  • Object: Maximum set of interface nodes that belong to the same set of

subdomains

6 / 24

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SLIDE 14

Premilinaries

  • Element-based (non-overlapping DD) distribution (+ limited ghost info)

Th T 1

h , T 2 h , T 3 h

˜ T 1

h

  • Gluing info based on objects
  • Object: Maximum set of interface nodes that belong to the same set of

subdomains

6 / 24

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SLIDE 15

Automatic hierarchical mesh generator

Classification of objects (vef’s at the next level) in 3D

  • Faces: Objects that belong to 2 subdomains
  • Edges: Objects that belong to more than 2 subdomains
  • Corners: Edges and faces with cardinality 1

7 / 24

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SLIDE 16

Coarser triangulation

  • Similar to FE triangulation object but wo/ reference element
  • Instead, aggregation info
  • bject level 1 = aggregation (vef’s level 0)

8 / 24

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SLIDE 17

Coarser FE space

  • On top of coarser triangulation, we create a FE-like functional space
  • DOFs on geometrical objects at the coarser level (as in FEs)
  • Aggregation info for DOFs (uα

1 = Fα(u1))

9 / 24

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SLIDE 18

Coarser FE space

  • On top of coarser triangulation, we create a FE-like functional space
  • DOFs on geometrical objects at the coarser level (as in FEs)
  • Aggregation info for DOFs (uα

1 = Fα(u1))

9 / 24

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SLIDE 19

Coarser FE space

  • On top of coarser triangulation, we create a FE-like functional space
  • DOFs on geometrical objects at the coarser level (as in FEs)
  • Aggregation info for DOFs (uα

1 = Fα(u1))

uα

1 =

1 #(p) X

p∈Eα

u1(p)

9 / 24

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SLIDE 20

Hierarchical FE spaces

  • The under-assembled space ¯

V0 = {v ∈ ˜ V0| continuous F1(v)}

  • ¯

V0 is a multiscale space

V0 ˜ V0 ¯ V0

  • Compute sol’on in V0 using ¯

V0 correction as preconditioner (multilevel precond)

  • BDDC DD preconditioner is a particular realization of ¯

V0 (corners/edges/faces)

10 / 24

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SLIDE 21

Hierarchical FE spaces

  • The under-assembled space ¯

V0 = {v ∈ ˜ V0| continuous F1(v)}

  • ¯

V0 is a multiscale space

V0 ˜ V0 ¯ V0

  • Compute sol’on in V0 using ¯

V0 correction as preconditioner (multilevel precond)

  • BDDC DD preconditioner is a particular realization of ¯

V0 (corners/edges/faces)

10 / 24

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SLIDE 22

Hierarchical FE spaces

  • The under-assembled space ¯

V0 = {v ∈ ˜ V0| continuous F1(v)}

  • ¯

V0 is a multiscale space

V0 ˜ V0 ¯ V0

  • Compute sol’on in V0 using ¯

V0 correction as preconditioner (multilevel precond)

  • BDDC DD preconditioner is a particular realization of ¯

V0 (corners/edges/faces)

10 / 24

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SLIDE 23

Hierarchical FE spaces

The under-assembled space ¯ V0 can be decomposed as [Dohrmann’03]:

  • Its bubble space ¯

V b

0 = {v ∈ ¯

V0|F(v) = 0}

  • The coarser FE space V1 = {v ∈ ¯

V0|v ⊥ ˜

A ¯

V b

0 }

F(u0) = 0

¯ V0 = ¯ V b ⊕ V1

11 / 24

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SLIDE 24

Hierarchical FE spaces

The under-assembled space ¯ V0 can be decomposed as [Dohrmann’03]:

  • Its bubble space ¯

V b

0 = {v ∈ ¯

V0|F(v) = 0}

  • The coarser FE space V1 = {v ∈ ¯

V0|v ⊥ ˜

A ¯

V b

0 }

F(u0) = 0

¯ V0 = ¯ V b ⊕ V1

11 / 24

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SLIDE 25

Coarse corner function

  • Compute via local problems a basis for V1 = {Φ1, . . . , Φnc}
  • Every Φ is a coarse shape function related to a coarse DoF

Circle domain partitioned into 9 subdomains V1 corner basis function

12 / 24

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SLIDE 26

Coarse edge function

  • Compute via local problems a basis for V1 = {Φ1, . . . , Φnc}
  • Every Φ is a coarse shape function related to a coarse DoF

Circle domain partitioned into 9 subdomains V1 edge basis function

13 / 24

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SLIDE 27

Multilevel/scale concurrency

The problem in ¯ V0 = V1 ⊕ V b

0 :

¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 can be decomposed as ¯ u0 = ¯ ub

0 + u1 (orthogonality V1 ⊥ ˜ A ¯

V b

0 )

ub

0 ∈ ¯

V b : a(ub

0 , v b 0 ) = (f0, v b 0 ) ∀v0 ∈ ¯

V b u1 ∈ V1 : a(u1, v1) = (f1, v1) ∀v1 ∈ V1

  • Bubble component is local to every subdomain (parallel)
  • Coarse global problem

14 / 24

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SLIDE 28

Multilevel/scale concurrency

The problem in ¯ V0 = V1 ⊕ V b

0 :

¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 can be decomposed as ¯ u0 = ¯ ub

0 + u1 (orthogonality V1 ⊥ ˜ A ¯

V b

0 )

ub

0 ∈ ¯

V b : a(ub

0 , v b 0 ) = (f0, v b 0 ) ∀v0 ∈ ¯

V b u1 ∈ V1 : a(u1, v1) = (f1, v1) ∀v1 ∈ V1

  • Bubble component is local to every subdomain (parallel)
  • Coarse global problem

14 / 24

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SLIDE 29

Multilevel/scale concurrency

The problem in ¯ V0 = V1 ⊕ V b

0 :

¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 can be decomposed as ¯ u0 = ¯ ub

0 + u1 (orthogonality V1 ⊥ ˜ A ¯

V b

0 )

ub

0 ∈ ¯

V b : a(ub

0 , v b 0 ) = (f0, v b 0 ) ∀v0 ∈ ¯

V b u1 ∈ V1 : a(u1, v1) = (f1, v1) ∀v1 ∈ V1

  • Bubble component is local to every subdomain (parallel)
  • Coarse global problem

14 / 24

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SLIDE 30

Multilevel/scale concurrency

The problem in ¯ V0 = V1 ⊕ V b

0 :

¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 can be decomposed as ¯ u0 = ¯ ub

0 + u1 (orthogonality V1 ⊥ ˜ A ¯

V b

0 )

ub

0 ∈ ¯

V b : a(ub

0 , v b 0 ) = (f0, v b 0 ) ∀v0 ∈ ¯

V b u1 ∈ V1 : a(u1, v1) = (f1, v1) ∀v1 ∈ V1

  • Bubble component is local to every subdomain (parallel)
  • Coarse global problem

Multilevel concurrency is BASIC for extreme scalability implementations

14 / 24

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SLIDE 31

Multilevel concurrency

P0 = P1 P2 t

  • L1 duties are fully parallel
  • L2 duties destroy scalability because
  • # L1 proc’s ∼ × 1000 # L2 proc’s
  • L2 problem size increases w/ number of proc’s

15 / 24

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SLIDE 32

Multilevel concurrency

P0 = P1 P2 t P3

  • Every processor has one level/scale duties
  • Idling dramatically reduced (energy-aware solvers)
  • Overlapped communications / computations among levels

15 / 24

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SLIDE 33

Multilevel concurrency

P0 = P1 P2 t P3

Inter-level overlapped bulk asynchronous (MPMD) im- plementation in FEMPAR

15 / 24

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SLIDE 34

FEMPAR implementation Multilevel extension straightforward (starting the alg’thm with V1 and level-1 mesh)

.....

c

  • r

e 1 c

  • r

e 2 c

  • r

e 3 c

  • r

e 4 c

  • r

e P 1 1st level MPI comm

..... .....

c

  • r

e 1 c

  • r

e 2 c

  • r

e P 2 2nd level MPI comm

.....

3rd level MPI comm c

  • r

e 1 parallel (distributed) global communication global communication

.....

time 16 / 24

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SLIDE 35

FEMPAR implementation Multilevel extension straightforward (starting the alg’thm with V1 and level-1 mesh) Extremely scalable implementation in FEMPAR:

  • Recursive (extensible to arbitrary # of levels)
  • Inter-level overlapped (bulk asynchronous)

16 / 24

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SLIDE 36

Outline

1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4 Conclusions

16 / 24

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SLIDE 37

BDDC preconditioning

BDDC preconditioner [Dohrmann’03, . . .]

  • Replace V0 by ¯

V0 (reduced continuity)

  • Define the injection I : ¯

V0 − → V0 (weight, comm and add)

  • Find ¯

u0 ∈ ¯ V0 such that: ¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 and obtain u = MBDDCr = EI¯ u0, where E is the harmonic extension operator (correct in the interior of subdomains) V0 ¯ V0

17 / 24

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SLIDE 38

BDDC preconditioning

BDDC preconditioner [Dohrmann’03, . . .]

  • Replace V0 by ¯

V0 (reduced continuity)

  • Define the injection I : ¯

V0 − → V0 (weight, comm and add)

  • Find ¯

u0 ∈ ¯ V0 such that: ¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 and obtain u = MBDDCr = EI¯ u0, where E is the harmonic extension operator (correct in the interior of subdomains) V0 I It ¯ V0

17 / 24

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SLIDE 39

BDDC preconditioning

BDDC preconditioner [Dohrmann’03, . . .]

  • Replace V0 by ¯

V0 (reduced continuity)

  • Define the injection I : ¯

V0 − → V0 (weight, comm and add)

  • Find ¯

u0 ∈ ¯ V0 such that: ¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 and obtain u = MBDDCr = EI¯ u0, where E is the harmonic extension operator (correct in the interior of subdomains) V0 I It ¯ V0

17 / 24

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SLIDE 40

BDDC preconditioning

BDDC preconditioner [Dohrmann’03, . . .]

  • Replace V0 by ¯

V0 (reduced continuity)

  • Define the injection I : ¯

V0 − → V0 (weight, comm and add)

  • Find ¯

u0 ∈ ¯ V0 such that: ¯ u0 ∈ ¯ V0 : a(¯ u0, ¯ v0) = (f , ¯ v0) ∀¯ v0 ∈ ¯ V0 and obtain u = MBDDCr = EI¯ u0, where E is the harmonic extension operator (correct in the interior of subdomains)

The application of MBDDC(·) can be im- plemented using the multilevel frame- work above

V0 I It ¯ V0

17 / 24

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SLIDE 41

Overlapping regions

  • Classify duties among levels
  • 3 overlapping regions (!)

Solve Ax = b w/ BDDC-PCG Precond’er set-up (MBDDC) call PCG(A,MBDDC,b,x0) PCG r0 := b − Ax0 z0 := M−1

BDDCr0

p0 := z0 for j = 0, . . . , till CONV do sj+1 = Apj . . . zj+1 := M−1

BDDCrj+1

. . . end for

18 / 24

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SLIDE 42

PCG-BDDC tasks

L1 MPI tasks L2 MPI tasks L3 MPI task Identify local coarse DoFs Gather coarse-grid DoFs Algorithm 1 (k ≡ iL1) Build G

A

(jL2 ) C

Algorithm 2 (k ≡ iL1) Identify local coarse DoFs Compute ΦiL1 Gather coarse-grid DoFs A

(iL1) C

← Φt

iL1 (−C T iL1 ΛiL1 )

Algorithm 1 (k ≡ iL2) Build GAC Gather A

(iL1) C

Re+Sy fact(GAC) Algorithm 3 (k ≡ iL1) A

(jL2) C

:= assemb(A

(iL1) C

) Algorithm 4 (k ≡ iL1) Algorithm 2 (k ≡ iL2) Compute ΦiL2 A

(iL2) C

← Φt

iL2 (−C T iL2 ΛiL2 )

Gather A

(iL2) C

Algorithm 3 (k ≡ iL2) AC := assemb(A

(iL2) C

) Num fact(AC) Gather r

(iL1) C

Algorithm 5 (k ≡ iL1) r

(jL2) C

:= assemb(r

(iL1) C

) Algorithm 4 (k ≡ iL2) Gather r

(iL2) 1

Algorithm 5 (k ≡ iL2) rC := assemb(r

(iL2) C

) Solve ACzC = rC Scatter zC into z

(iL2) C

Algorithm 6 (k ≡ iL2) Scatter z

(jL2) C

into z

(iL1) C

Algorithm 6 (k ≡ iL1) Algorithm 1

Re+Sy fact(GA(k)

F

) Re+Sy fact(GA(k)

II

)

Algorithm 2

Num fact((Ab

0 )(k))

Algorithm 3

Num fact(A(k)

II )

Algorithm 4

δ(k)

I

← (A(k)

II )−1r(k) I

r(k)

Γ

← r(k)

Γ

− A(k)

ΓI δ(k) I

r(k) ← It

kr

Algorithm 5

Solve (Ab

0 )(k)

  • t

s(k)

F

λ

  • =
  • r(k)
  • Algorithm 6

s(k)

C ← Φiz(k) C

z(k) ← Ii(s(k)

F + s(k) C )

z(k)

I

← −(A(k)

II )−1A(k) IΓ z(k) Γ

z(k)

I

← z(k)

I

+ δ(k)

I

19 / 24

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SLIDE 43

Interlevel load balance

Goal: strike a balance such that blue/red areas are kept below green ones!

0.2 0.4 0.6 0.8 1 512 4K 8K 13.8K 21.9K 27K 32K 46.6K #cores Weak scaling for MLBDDC(cef) solver with 1K FEs/core 3-lev BDDC (Heavy 3rd Level) 3-lev BDDC (Heavy 2nd Level) 4-lev BDDC (Well Balanced)

20 / 24

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SLIDE 44

Weak scaling 3-lev BDDC(ce) solver

3D Laplacian problem on IBM BG/Q (JUQUEEN@JSC) 16 MPI tasks/compute node, 1 OpenMP thread/MPI task

10 20 30 40 50 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores Weak scaling for MLBDDC(ce) solver 3-lev H1/h1=20 H2/h2=7 4-lev H1/h1=20 H2/h2=3 H3/h3=3 3-lev H1/h1=25 H2/h2=7 4-lev H1/h1=25 H2/h2=3 H3/h3=3 3-lev H1/h1=30 H2/h2=7 3-lev H1/h1=40 H2/h2=7 5 10 15 20 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores 30 40 50 60 70 80 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores Weak scaling for MLBDDC(ce) solver 3-lev H1/h1=40 H2/h2=7

#PCG iterations Total time (secs.)

Experiment set-up Lev. # MPI tasks FEs/core 1st 42.8K 74.1K 117.6K 175.6K 250K 343K 456.5K 203/253/303/403 2nd 125 216 343 512 729 1000 1331 73 3rd 1 1 1 1 1 1 1 n/a

21 / 24

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SLIDE 45

Weak scaling 3-lev BDDC(ce) solver

3D Linear Elasticity problem on IBM BG/Q (JUQUEEN@JSC) 16 MPI tasks/compute node, 1 OpenMP thread/MPI task

10 20 30 40 50 60 70 80 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores Weak scaling for MLBDDC(ce) solver 3-lev H1/h1=15 H2/h2=7 3-lev H1/h1=20 H2/h2=7 3-lev H1/h1=25 H2/h2=7 10 20 30 40 50 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores 3-lev H1/h1=15 H2/h2=7 3-lev H1/h1=20 H2/h2=7 60 80 100 120 140 160 180 2.7K 42.8K 117.6K 175.6K 250K 343K 458K #cores Weak scaling for MLBDDC(ce) solver 3-lev H1/h1=25 H2/h2=7

#PCG iterations Total time (secs.)

Experiment set-up Lev. # MPI tasks FEs/core 1st 42.8K 74.1K 117.6K 175.6K 250K 343K 456.5K 153/203/253 2nd 125 216 343 512 729 1000 1331 73 3rd 1 1 1 1 1 1 1 n/a

22 / 24

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SLIDE 46

Weak scaling 4-lev BDDC(ce)

3D Laplacian problem on IBM BG/Q (JUQUEEN@JSC) 64 MPI tasks/compute node, 1 OpenMP thread/MPI task

10 20 30 40 50 60 46.6K 216K 373.2K 592.7K 884.7K 1.26M 1.73M 12.1K 56K 96.8K 153.7K 229.5K 326.8K 448.3K # PCG iterations #subdomains Weak scaling for 4-level BDDC(ce) solver with H2/h2=4, H3/h3=3 #cores H1/h1=10 H1/h1=20 H1/h1=25

1 2 3 4 5 6 46.6K 216K373.2K 592.7K 884.7K 1.26M 1.73M 12.1K 56K 96.8K 153.7K 229.5K 326.8K 448.3K #subdomains H1/h1=10 H1/h1=20 H1/h1=25 10 15 20 25 30 35 40 46.6K 216K373.2K 592.7K 884.7K 1.26M 1.73M 12.1K 56K 96.8K 153.7K 229.5K 326.8K 448.3K Weak scaling for 4-level BDDC(ce) solver #cores

Total time (secs.)

Lev. # MPI tasks FEs/core 1st 46.7K 110.6K 216K 373.2K 592.7K 884.7K 1.26M 103/203/253 2nd 729 1.73K 3.38K 5.83K 9.26K 13.8K 19.7K 43 3rd 27 64 125 216 343 512 729 33 4th 1 1 1 1 1 1 1 n/a

23 / 24

slide-47
SLIDE 47

Outline

1 Motivation 2 Multilevel framework 3 Multilevel linear solvers 4 Conclusions

23 / 24

slide-48
SLIDE 48

Conclusion

  • Extremely scalable implementation (MLBDDC)
  • Fully-distributed and communicator-aware
  • Interlevel-overlapped (bulk-asynchronous)
  • Recursive (extensible to arbitrary # levels)
  • Remarkable scalability
  • 3D Laplacian and Linear Elasticity PDEs
  • 3/4 levels are sufficient to (efficiently) scale till

full JUQUEEN

  • More levels probably needed in the future

24 / 24

slide-49
SLIDE 49

Conclusion

  • Extremely scalable implementation (MLBDDC)
  • Fully-distributed and communicator-aware
  • Interlevel-overlapped (bulk-asynchronous)
  • Recursive (extensible to arbitrary # levels)
  • Remarkable scalability
  • 3D Laplacian and Linear Elasticity PDEs
  • 3/4 levels are sufficient to (efficiently) scale till

full JUQUEEN

  • More levels probably needed in the future

24 / 24

slide-50
SLIDE 50

Conclusion

  • Extremely scalable implementation (MLBDDC)
  • Fully-distributed and communicator-aware
  • Interlevel-overlapped (bulk-asynchronous)
  • Recursive (extensible to arbitrary # levels)
  • Remarkable scalability
  • 3D Laplacian and Linear Elasticity PDEs
  • 3/4 levels are sufficient to (efficiently) scale till

full JUQUEEN

  • More levels probably needed in the future

Future work:

  • Unstructured mesh weak scalability analyses (technical aspects, mesh

generation)

  • Include adaptive selection of coarse DOFs (not so important in

hydrodynamics)

24 / 24

slide-51
SLIDE 51

Farewell

Thank you!

SB, A. F. Martín and J. Principe. Multilevel Balancing Domain Decomposition at Extreme Scales. Submitted, 2015.

Work funded by the European Research Council under:

  • Starting Grant 258443 - COMFUS: Computational Methods

for Fusion Technology

  • Proof of Concept Grant 640957 - FEXFEM: On a free open

source extreme scale finite element software

2007 - 2012

European Research Council

Years

  • f excellent IDEAS

24 / 24