Parallel Iterative Solvers with Preconditioning in the Post- Moore Era Kengo Nakajima Information Technology Center, The University of Tokyo First International Workshop on Deepening Performance Models for Automatic Tuning (DPMAT) September 7, 2016, Nagoya University
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Parallel Iterative Solvers with Preconditioning in the Post-
Moore EraKengo Nakajima
Information Technology Center, The University of Tokyo
First International Workshop on Deepening Performance Models for Automatic Tuning (DPMAT)September 7, 2016, Nagoya University
2
• Both of convergence (robustness) and efficiency (single/parallel) are important
• Power-aware Methods– Approximate Computing, Power Management, FPGA
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Hierarchical Methods for Hiding Latency6
hCGA in Parallel Multigrid
5.0
7.5
10.0
12.5
15.0
100 1000 10000 100000
sec.
CORE#
CGAhCGA
x1.61
Groundwater Flow Simulation with up to 4,096 nodes on Fujitsu FX10 (GMG-CG) up to 17,179,869,184 meshes (643 meshes/core) [KN ICPADS 2014]
Parallel in Space/Time (PiST)PiST approach is suitable for the Post-Moore Systems with a complex and deeply hierarchical network that causes large latency.
CGA hCGA
[R.D.Falgout et al. SIAM/SISC 2014]
Comparison between PiST and “Time Stepping” for Transient Poisson EquationsEffective if processor# is VERY large
3D: 333 x 40978 proc’s in space dir.
App’s & Alg’s in Post-Moore Era• Compute Intensity -> Data Movement Intensity
– It is very important and helpful for the convergence of BDA and HPC to think about algorithms and applications in the Post Moore Era.
• Implicit scheme strikes back !: but not straightforward• Hierarchical Methods for Hiding Latency
– Hierarchical Coarse Grid Aggregation (hCGA) in MG– Parallel in Space/Time (PiST)
• Comm./Synch. Avoiding/Reducing Algorithms– Network latency is already a big bottleneck for parallel
– Demmel, Hoemmen, Mohiyuddin etc. (UC Berkeley)• s-step method
– Just one P2P communication for each Mat-Vec during siterations. Convergence may become unstable for large s.
• Communication Avoiding ILU0 (CA-ILU0) [Moufawad & Grigori, 2013]– First attempt to CA preconditioning– Nested dissection reordering for limited geometries (2D FDM)
• Generally, it is difficult to apply Matrix Powers Kernel to preconditioned iterative solvers
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Comm. Avoiding Krylov Iterative Methods using “Matrix Powers Kernel”
ISS-2013 11
Avoiding Communication in Sparse Matrix Computations. James Demmel, Mark Hoemmen, Marghoob Mohiyuddin, and Katherine Yelick. , 2008 IPDPS
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• Communication/Synchronization Avoiding/Reducing in Krylov Iterative Solvers
Hiding Overhead by Collective Comm. in Krylov Iterative Solvers
• Dot Products in Krylov Iterative Solvers– MPI_Allreduce: Collective Communications– Large overhead with many nodes
• Pipelined CG [Ghysels et al. 2014]– Utilization of asynchronous collective communications (e.g.
MPI_Iallreduce) supported in MPI-3 for hiding such overhead.– Algorithm is kept, but order of computations is changed– [Reference] P. Ghysels et al., Hiding global synchronization
latency in the preconditioned Conjugate Gradient algorithm, Parallel Computing 40, 2014
– When I visited LBNL in September 2013, Dr. Ghysels asked me to evaluate his idea in my parallel multigrid solvers
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4 Algorithms [Ghysels et al. 2014]• Alg.1 Original Preconditioned CG• Alg.2 Chronopoulos/Gear
– 2 dot products are combined in a single reduction• Alg.3 Pipelined CG (MPI_Iallreduce)• Alg.4 Gropp’s asynchronous CG (MPI_Iallreduce)
• Algorithm itself is not different from the original one– Recurrence Relations: 漸化式
– Order of computation changed -> Rounding errors are propagated differently
• Convergence may be affected (not happened in my case)• update of r= b-Ax needed at every 50 iterations (original paper)
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Original Preconditioned CG (Alg.1)Original Preconditioned CG (Alg.1)
iiiiii
iiiii
iiii
ii
srAprApAxbAxbr
pxxAps
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1
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Chronopoulos/Gear CG (Alg.2)2 dot products are combined into a single reduction
Chronopoulos/Gear CG (Alg.2)
iiiiii
iiiii
iiii
iiiiiiiiii
srAprApAxbAxbr
pxxApspupsAus
11
1
1,
• 2 dot products combined
• si=Api is not computed explicitly: by recurrence
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Pipelined Chronopoulos/Gear(No Preconditioning)
Pipelined Chronopoulos/Gear (No Precond.)
iiiiii
iii
iiiiiiiiii
iiiiiiii
iiiii
srAprrAAwq
zAwpAAszAsAwAs
AswwAsArArArAuwru
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21
11
,
• Global synchronization of dot products are overlapped with SpMV
• Global synchronization of dot products are overlapped with SpMV and Preconditioning
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Gropp’s Asynchronous CG (Alg.4)Smaller Computations than Alg.3
Gropp’s Asynchronous CG (Alg.4)
• Definition of is different from that of Alg.3
W. Gropp, Update on Libraries for Blue Waters.http://jointlab-pc.ncsa.illinois.edu/events/workshop3/pdf/presentations/Gropp-Update-on-Libraries.pdfPresentation Material (not a paper, article)
• Global synchronization of dot products are overlapped with SpMV and Preconditioning
do i= N, 1, -1SW = 0.0d0do k= indexU(i), indexU(i-1)+1, -1SW= SW + AU(k) * W(itemU(k),Z)
enddoW(i,Z)= W(i,Z) – SW / D(i)
enddo!C===
Ignoring effects of external points for preconditioning Block-Jacobi Localized
Preconditioning WEAKER than original
SGS/SSOR More PE’s, more iterations
Parallel FEM 3D-2 26
Localized SGS/SSORPreconditioning
A1 2 3 4 5 6PE#1
PE#2
PE#3
PE#4
1 2 3
4 5 6
Considered :
Ignored :
Parallel FEM 3D-2 2727
Overlapped Additive Schwartz Domain Decomposition MethodStabilization of Localized Preconditioning: ASDD
Global Operation
Local Operation
Global Nesting Correction: Repeating -> Stable
1 2
rMz
222111
11 ,
rMzrMz
)( 111111111111
nnnn zMzMrMzz
)( 111122222222
nnnn zMzMrMzz
Parallel FEM 3D-2 2828
Overlapped Additive Schwartz Domain Decomposition MethodEffect of additive Schwartz domain decomposition for solid mechanics example example with 3x443 DOF on Hitachi SR2201, Number of ASDD cycle/iteration= 1, = 10-8
PE # Iter. # Sec. Speed Up Iter.# Sec. Speed Up1 204 233.7 - 144 325.6 -2 253 143.6 1.63 144 163.1 1.994 259 74.3 3.15 145 82.4 3.958 264 36.8 6.36 146 39.7 8.21
Results: Speed-Up: SmallPerformance of 2 nodes of Flat MPI = 64.0
(4 sockets, 64 cores)
Flat MPI Hybrid
Alg.1 Original PCGAlg.2 Chronopoulos/GearAlg.3 Pipelined CGAlg.4 Gropp’s CG
0
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p
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Alg.1Alg.2Alg.3Alg.4Ideal
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Alg.1Alg.2Alg.3Alg.4Ideal
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Alg.1Alg.2Alg.3Alg.4Ideal
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p
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Alg.1Alg.2Alg.3Alg.4Ideal
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Results: Speed-Up: MediumPerformance of 2 nodes of Flat MPI = 64.0
(4 sockets, 64 cores)
Flat MPI Hybrid
Alg.1 Original PCGAlg.2 Chronopoulos/GearAlg.3 Pipelined CGAlg.4 Gropp’s CG
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Preliminary Results on IVB Cluster96x80x64 (491,520) nodes, 1,474,560 DOFFlat MPI using up to 64 nodes (1,280 cores)Flat MPI worked well in this caseAt 64 nodes, problem size per core is equal to that of “small” case
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Spee
d-U
p
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Alg.1 Alg.2Alg.3 Alg.4Ideal
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0 160 320 480 640 800 960 1120 1280
Rel
ativ
e Pe
rfor
ance
(%)
CORE#
Alg.2 Alg.3 Alg.4
Speed-Up (20-1,280 cores) Relative Performance to Alg.1 (Original)
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Allreduce vs. Iallreduce for HybridPerformance of 2 nodes of Flat MPI = 64.0
(4 sockets, 64 cores)Alg.1 Original PCGAlg.3 Pipelined CGAlg.4 Gropp’s CG
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0 2048 4096 6144 8192 10240 12288
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Alg.1Alg.3-IARAlg.4-IARIdealAlg.3-ARAlg.4-AR
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Alg.1Alg.3-IARAlg.4-IARIdealAlg.3-ARAlg.4-AR
Small Medium
IAR: MPI_IallreduceAR : MPI_Allreduce
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Hybrid vs. Flat MPI for Alg.4
0.00
1.00
2.00
3.00
4.00
0 2048 4096 6144 8192 10240 12288
Hyb
rid/F
lat M
PI R
atio
CORE#
Alg.4-SmallAlg.4-Medium
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Results on 768 nodes (12,288 cores) of Fujitsu FX10 (Oakleaf-FX)
MPI-3 is not optimizedFX100 has special HW for communication
0.00
0.20
0.40
0.60
0.80
1.00
1.20
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1.60
1.80
Alg.1 Alg.2 Alg.3-IAR Alg.3-AR Alg.4-IAR Alg.4-AR
sec.
Algorithms
Small: Flat MPISmall: Hybrid
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• Communication/Synchronization Avoiding/Reducing in Krylov Iterative Solvers
Mat-Vec operations (SpMV)• Renumbering: ■⇒■• Communications of info. on
external meshes• Computation of ■ BEFORE
completion of comm. (comm.-comp. overlapping)
• Synchronization of communications
• Computation of ■
Comm.-Comp. OverlappingCC-Overlapping 41
Internal Meshes
External (HALO) Meshes
Internal Meshes on Boundary’s
call MPI_Isendcall MPI_Irecv
do i= 1, Ninn(calculations)
enddo
call MPI_Waitall
do i= Ninn+1, Nall(calculationas)
enddo
With Renumbering
Comm.-Comp. Overlappingfor SpMV
• No effects on SpMV (will be shown later)• We need certain amount of communications• Larger communications mean larger computations
– Ratio of communication overhead is small …– Communication time itself is not so large
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OpenMP: Loop Scheduling!$omp parallel do schedule (kind, [chunk]) !$omp do schedule (kind, [chunk])
#pragma parallel for schedule (kind, [chunk]) #pragma for schedule (kind, [chunk])
Kind Description
staticDivide the loop into equal-sized chunks or as equal as possible in the case where the number of loop iterations is not evenly divisible by the number of threads multiplied by the chunk size. By default, chunk size is loop_count/number_of_threads.Set chunk to 1 to interleave the iterations.
dynamic
Use the internal work queue to give a chunk-sized block of loop iterations to each thread. When a thread is finished, it retrieves the next block of loop iterations from the top of the work queue. By default, the chunk size is 1. Be careful when using this scheduling type because of the extra overhead involved.
guidedSimilar to dynamic scheduling, but the chunk size starts off large and decreases to better handle load imbalance between iterations. The optional chunk parameter specifies them minimum size chunk to use. By default the chunk size is approximately loop_count/number_of_threads.
autoWhen schedule (auto) is specified, the decision regarding scheduling is delegated to the compiler. The programmer gives the compiler the freedom to choose any possible mapping of iterations to threads in the team.
runtimeUses the OMP_schedule environment variable to specify which one of the three loop-scheduling types should be used. OMP_SCHEDULE is a string formatted exactly the same as would appear on the parallel construct.
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Strategy [Idomura et al. 2014]• “dynamic”• “!$omp master~!$omp end master”
!$omp master Communication is done by the master thread (#0)!C!C– Send & Recv.(…)
call MPI_WAITALL (2*NEIBPETOT, req1, sta1, ierr)!$omp end master
!C The master thread can join computing of internal!C-- Pure Inner Nodes nodes after the completion of communication
!$omp do schedule (dynamic,200) Chunk Size= 200do j= 1, Ninn(…)
enddo!C!C-- Boundary Nodes Computing for boundary nodes are by all threads
!$omp do default: !$omp do schedule (static)do j= Ninn+1, N(…)
enddo
!$omp end parallel
Idomura, Y. et al., Communication-overlap techniques for improved strong scaling of gyrokinetic Eulerian code beyond 100k cores on the K-computer, Int. J. HPC Appl. 28, 73-86, 2014
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Block Diagonal CGsec./iteration
(1/2)
CC-Overlapping
(4.00)
(2.00)
0.00
2.00
4.00
6.00
8.00
10.00
Overlap 50 100 200 300 400 500
Spee
d-up
(%)
FX10-128-S RB-128-S KNSC-128-S
• Hybrid• Small:1003 nodes/proc.• Large:2003 nodes/proc.• Overlap: Classical Method• Number: Chunk Size• Difference from the