Distributed Dense Linear Algebra on Heterogeneous Architectures. George Bosilca
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1 Distributed Dense Linear Algebra on Heterogeneous Architectures George Bosilca Centraro, Italy June 2010
2 Factors that Necessitate to Redesign of Our Software» Steepness of the ascent from terascale to petascale to exascale» Extreme parallelism and hybrid design» Preparing for million/billion way parallelism» Tightening memory/bandwidth bottleneck» Limits on power/clock speed implication on multicore» Reducing communication will become much more intense» Memory per core changes, byte-to-flop ratio will change» Necessary Fault Tolerance» MTTF will drop» Checkpoint/restart has limitations Software infrastructure does not exist today 100,000 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 Average Number of Cores Per Supercomputer for Top20 Systems
3 Linpack Evolution Software/Algorithms follow hardware evolution in time LINPACK (70 s) (Vector operations) Rely on - Level-1 BLAS operations
4 Linpack (R)Evolution Software/Algorithms follow hardware evolution in time LINPACK (70 s) (Vector operations) Rely on - Level-1 BLAS operations LAPACK (80 s) (Blocking, cache friendly) Rely on - Level-3 BLAS operations
5 Linpack Evolution Software/Algorithms follow hardware evolution in time LINPACK (70 s) (Vector operations) Rely on - Level-1 BLAS operations LAPACK (80 s) (Blocking, cache friendly) ScaLAPACK (90 s) (Distributed Memory) Rely on - Level-3 BLAS operations Rely on - PBLAS Mess Passing
6 Linpack Evolution Software/Algorithms follow hardware evolution in time LINPACK (70 s) (Vector operations) Rely on - Level-1 BLAS operations LAPACK (80 s) (Blocking, cache friendly) ScaLAPACK (90 s) (Distributed Memory) PLASMA (00 s) New Algorithms (many-core friendly) Rely on - Level-3 BLAS operations Rely on - PBLAS Mess Passing Rely on - a DAG/scheduler - block data layout - some extra kernels
7 Linpack Evolution Software/Algorithms follow hardware evolution in time PLASMA (00 s) New Algorithms (many-core friendly) Rely on - a DAG/scheduler - block data layout - some extra kernels Those new algorithms - have a very low granularity, they scale very well (multicore, petascale computing, ) - removes of dependencies among the tasks, (multicore, distributed computing) - avoid latency (distributed computing, out-of-core) - rely on fast kernels Those new algorithms need new kernels and rely on efficient scheduling algorithms.
8 The algorithmic challenge Tiled Column-major / columnmajor Flat (no indirection)» Asynchronicity» Avoid fork-join (Bulk sync design)» Dynamic Scheduling» Out of order execution» Fine Granularity» Independent block operations» Locality of Reference» Data storage Block Data Layout
9 LAPACK LU Step 1 Step 2 Step 3 Step 4...» Fork-join, bulk synchronous processing 9
10 » Expose the intrinsic parallelism: break into smaller tasks and remove dependencies
11 Expose the intrinsic parallelism» The three amigos» Example of QR factorization» 4 basic kernels» LU identical to QR except using different kernels» Cholesky slightly different due to the matrix symmetry
12 LU DAG representation
13 A quite simple problem» We would generate the DAG, find the critical path and execute it.» DAG too large to generate ahead of time» Not explicitly generate» Dynamically generate the DAG as we go» Machines will have large number of cores in a distributed fashion» Will have to engage in message passing» Distributed management» Locally have a run time system
14 What is PLASMA? Dense linear algebra software library Linear systems & least squares (LU, Cholesky, QR/LQ) Eigenvalues & singular values Multicore processors Multi-socket multi-core / Shared memory systems NUMA systems What is next: GPU acceleration MAGMA Distributed Memory DAGuE / DPLASMA
15 If the current architectures
16 PLASMA Performance (Cholesky, 48 cores) 250 Cholesky Performance (double prec.) PLASMA 200 MKL 11.0 LAPACK 150 Gflop/s Size 16
17 PLASMA Performance (LU, 48 Cores) LU Performance (double prec.) PLASMA MKL 11.0 LAPACK 120 Gflop/s Size
18 PLASMA Performance (QR, 48 cores) Cholesky Performance (double prec.) PLASMA MKL 11.0 LAPACK 200 Gflop/s Size
19 If the current architectures
20 NVIDIA Tesla C2050 (Fermi), GF100 Chip» Marketing truth» 14 processing cores (448 ALUs)» 32 CUDA Cores (SIMD functional units) per core» 1 mul-add (2 flops) per ALU (2 flops/cycle)» Best case theoretically: 448 mul-adds» 1.15 GHz clock» 14 * 32 * 2 * 1.15 = 1.03 Tflop/s peak» All this is single precision» Double precision is half this rate, 515 Gflop/s» Dense Linear Algebra truth» 144 GB/s bus, 3 GB memory» In SP SGEMM performance 580 Gflop/s» In DP DGEMM performance 300 Gflop/s» Power: 247 W» Interface PCIex16 Processing Core
21 Cholesky with Multiple GPUs hybrid DPOTRF (4 Opteron 1.8GHz and 4 GPU TESLA C GHz) CPUs-4GPUs 3CPUs-3GPUs 2CPUs-2GPUs 1CPU-1GPU 200 Gflop/s Matrix Sizes 21
22 MAGMA Software " Available through MAGMA's homepage " Included are the 3 one-sided matrix factorizations " Iterative Refinement Algorithm (Mixed Precision) " Standard (LAPACK) data layout and accuracy " Two LAPACK-style interfaces» CPU interface: both input and output are on the CPU» GPU interface: both input and output are on the GPU
23 Current Architectures
24
25 DPLASMA / DAGuE» DAGuE: the runtime» Deploy a DAG on a heterogeneous distributed environment» Architecture aware» Minimize data movements (in and out the node)» Enforce data locality (cache / NUMA / GPU)» Move the data across nodes» DPLASMA: a algebraic description of a DAG» Define the data distribution» Describe the algorithm at a high level» A powerful description language (?)
26 Kernels description
27 Algebraic dependencies description Execution space Parallel partitioning Parameters Priority
28 Cholesky (4x4)
29 Heterogeneous multi-gpu
30 Cholesky (problem size) GFlop/s Matrix size (N) Theoretical peak GEMM peak DPLASMA DAGuE (NB=340) DSBP (NB=340) ScaLAPACK (NB=120) Griffon : 81 nodes, 648 cores, Infiniband 20Gbs
31 Cholesky (weak scalability) Theoretical peak GEMM peak DPLASMA DAGuE (NB=340) DSBP (NB=340) ScaLAPACK (NB=120) 4000 GFlop/s Number of cores
32 Cholesky (strong scalability) Theoretical peak GEMM peak DPLASMA DAGuE (NB=340) DSBP (NB=340) ScaLAPACK (NB=120) 5000 GFlop/s Number of cores
33 Composability» Totally remove the need for synchronization» Follow the data flow between algebraic description of DAGs» Shorten the time to completion
34 Composability
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