Accelerating linear algebra computations with hybrid GPU-multicore systems.
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1 Accelerating linear algebra computations with hybrid GPU-multicore systems. Marc Baboulin INRIA/Université Paris-Sud joint work with Jack Dongarra (University of Tennessee and Oak Ridge National Laboratory) and Stanimire Tomov (University of Tennessee) 4th Workshop INRIA-Illinois 11/23/2010 Marc Baboulin 1/34 Accelerating linear algebra computations
2 General framework How to speed up numerical simulations? Exploit advances in hardware (e.g multicore, GPUs, FPGAs, Cell), manage to use hardware efficiently for HPC applications Better numerical methods Impact on numerical libraries LAPACK, ScaLAPACK, sparse solvers, iterative solvers...have to be rethought and rewritten Need for fast Dense Linear Algebra (DLA) kernels in scientific calculations Marc Baboulin 2/34 Accelerating linear algebra computations
3 Outline 1 Taking advantage of new parallel architectures Marc Baboulin 3/34 Accelerating linear algebra computations
4 Outline 1 Taking advantage of new parallel architectures 2 Marc Baboulin 4/34 Accelerating linear algebra computations
5 Outline 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 5/34 Accelerating linear algebra computations
6 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 6/34 Accelerating linear algebra computations
7 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 7/34 Accelerating linear algebra computations
8 Hardware to software trends Processor speed improves 59% / year but memory bandwidth only by 23%, latency by 5.5% Marc Baboulin 8/34 Accelerating linear algebra computations
9 Hardware to software trends Marc Baboulin 9/34 Accelerating linear algebra computations
10 Motivation for heterogeneity-aware algorithms GPUs evolution: applications far beyond graphics, high bandwidth, programmability (CUDA), memory hierarchy, double precision arithmetic... Architectural trends have moved towards heterogeneous (CPU+GPU) designs Fully exploit the computational power that each of the hybrid components offers Need for linear algebra routines for hybrid systems: there is no self-contained library like LAPACK Marc Baboulin 10/34 Accelerating linear algebra computations
11 Designing algorithms for Multicore+GPU Represent LAPACK algorithms as a collection of BLAS-based tasks and dependencies among them rely on high performance of (CU)BLAS Abstract us from specificities in programming a GPU Properly schedule the tasks execution over the multicore and the GPU MAGMA: Matrix Algebra on GPU and Multicore Architectures DLA library for heterogeneous/hybrid architectures starting with current Multicore+GPU systems LAPACK-style interface U. Tennessee, U. California Berkeley, INRIA Marc Baboulin 11/34 Accelerating linear algebra computations
12 Task splitting and scheduling Algorithms as Directed Acyclic Graph (DAG) (small tasks/tiles for multicore) Marc Baboulin 12/34 Accelerating linear algebra computations
13 Task splitting and scheduling DAGs for hybrid systems (both small and large tasks) Marc Baboulin 13/34 Accelerating linear algebra computations
14 Principles of hybrid implementation BLAS-level parallelism where the matrix resides on the GPU (BLAS calls replaced by CUBLAS) Offload to the CPU small kernels that are inefficient for the GPU Use asynchronism between CPU and GPU whenever possible More details in [ Dongarra, Tomov, Baboulin, 2010 ] Marc Baboulin 14/34 Accelerating linear algebra computations
15 Example: Cholesky factorization Marc Baboulin 15/34 Accelerating linear algebra computations
16 LU factorization in double precision 240 FERMI MAGMA FERMI ASM ISTANBUL PLASMA ISTANBUL MKL GTX280 MAGMA GFlop/s FERMI Tesla C2050: 448 CUDA 1.15GHz SP/DP peak is 1030 / 515 GFlop/s ISTANBUL AMD 8 socket 6 core (48 SP/DP peak is 1075 / 538 GFlop/s Matrix Size Marc Baboulin 16/34 Accelerating linear algebra computations
17 Time breakdown for MAGMA QR (single precision) Intel Xeon (2 x GHz) - GeForce GTX 280 ( GHz). Marc Baboulin 17/34 Accelerating linear algebra computations
18 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 18/34 Accelerating linear algebra computations
19 Marc Baboulin 19/34 Accelerating linear algebra computations
20 Bulk of the computation in 32-bit arithmetic Postprocess the 32-bit solution by refining it into a solution that is 64-bit accurate Can be performed on the GPU Problem must be not ill-conditioned Software details in: M. Baboulin, A. Buttari, J. Dongarra, J. Kurzak, J. Langou, J. Langou, P. Luszczek, S. Tomov, Accelerating scientific computations with mixed precision algorithms. Computer Physics Communications, Vol. 180, No 12, pp (2009). Marc Baboulin 20/34 Accelerating linear algebra computations
21 Example of the Cholesky factorization 1: LL T A (ε s ) O(n 3 ) 2: solve Ly = b (ε s ) O(n 2 ) 3: solve L T x 0 = y (ε s ) O(n 2 ) do k = 1, 2,... 4: r k b Ax k 1 (ε d ) 5: solve Ly = r k (ε s ) 6: solve L T z k = y (ε s ) 7: x k x k 1 + z k (ε d ) check convergence done Marc Baboulin 21/34 Accelerating linear algebra computations
22 Mixed precision Cholesky factorization Solving Ax = b in DP accuracy, A is SPD (Performance on an Intel 2.33 GHz + NVIDIA GeForce GTX 280) Gflop/s SP Factorization Mixed Prec. Solver CPU Iterations Mixed Prec. Solver GPU Iterations DP Solver Matrix Size x 1,000 Slide 15 / 13 Marc Baboulin 22/34 Accelerating linear algebra computations
23 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 23/34 Accelerating linear algebra computations
24 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 24/34 Accelerating linear algebra computations
25 The issue of pivoting in linear systems General square system Ax = b, solved by Gaussian Elimination (GE) We interchange rows: partial pivoting (PP) stability Factorization PA = LU, where P permutation matrix Partial pivoting implemented in LAPACK, matlab... No floating point operation in pivoting but it involves irregular movement of data Communication overhead due to pivoting: O(n 2 ) comparisons, for some architectures (multicore, GPUs), up to 50% of the global computational time Marc Baboulin 25/34 Accelerating linear algebra computations
26 Other approaches Communication avoiding algorithms: L. Grigori, J. Demmel, and H. Xiang, Communication avoiding Gaussian elimination Supercomupting 2008 proceedings. J. Demmel, L. Grigori, M. F. Hoemmen, and J. Langou, Communication-optimal parallel and sequential QR and LU factorizations, In review, SISC. Minimize the number of messages exchanged during the panel factorization, stable in practice. GPU algorithms: V. Volkov, J. Demmel, LU, QR, Cholesky factorizations using vector capabilities of GPUs, Lapack Working note 204. Reduce the pivoting overhead from 56% to 1-10% by using innovative data structure. Marc Baboulin 26/34 Accelerating linear algebra computations
27 Random butterfly transformation (RBT) [ Parker,1995 ] proposed to make the matrix sufficiently random so that, with probability close to 1, pivoting is not needed Precondition A with random matrices: UAV to solve Ax = b, we instead solve (UAV )y = Ub followed by x = Vy Random matrices U and V are chosen among a particular class of matrices ( ) called butterfly matrices which are of P Q the form. where P, Q, R and S are diagonal R S n/2 n/2 matrices. Marc Baboulin 27/34 Accelerating linear algebra computations
28 Random butterfly transformation (RBT) Method:LU with no pivoting on a preconditioned matrix The preconditioning is cheap (O(n 2 ) operations) We do not pivot (RBT NP) or just within the first few rows of the panel (RBT LP) we have a fully BLAS 3 algorithm RBT may require some steps of iterative refinement in the working precision We take advantage of the GPU for all these calculations (preconditioning, factorization in SP, iterative refinement) More details in [ Baboulin, Dongarra, Tomov, 2008 ] Marc Baboulin 28/34 Accelerating linear algebra computations
29 Outline Taking advantage of new parallel architectures 1 Taking advantage of new parallel architectures 2 3 Marc Baboulin 29/34 Accelerating linear algebra computations
30 Accuracy of RBT Marc Baboulin 30/34 Accelerating linear algebra computations
31 Hybrid RBT LU factorization Load splitting for a hybrid LU factorization (8 cores+gpu) Marc Baboulin 31/34 Accelerating linear algebra computations
32 Performance (DP) Performance of RBT LU factorization (double precision) Intel Xeon (2 x GHz), GeForce GTX 280 ( GHz). Marc Baboulin 32/34 Accelerating linear algebra computations
33 Hybrid CPU/GPU library available (MAGMA 0.2) with main linear system solvers, including mixed precision iterative refinement More details at Randomization is very promising for accelerating linear algebra computations on multicore and/or GPU architectures Some ongoing work: -Hybrid implementation for SVD and eigensolvers -Apply statistical techniques to estimate condition number for very big problems Starting collaboration with Wen-mei Hwu (UIUC) and student Liwen Chang Marc Baboulin 33/34 Accelerating linear algebra computations
34 Some references for this talk [1] S. Tomov, J. Dongarra, M. Baboulin, Towards dense linear algebra for hybrid GPU accelerated manycore systems. Parallel Computing, Vol. 36, No 5&6, pp (2010). [2] M. Baboulin, A. Buttari, J. Dongarra, J. Kurzak, J. Langou, P. Luszczek, S. Tomov, Accelerating scientific computations with mixed precision algorithms. Computer Physics Communications, Vol. 180, No 12, pp (2009). [3] S. Tomov, J. Dongarra, Accelerating the reduction to upper-hessenberg form through hybrid GPU-based computing. LAPACK Working Note 219 (2009). [4] M. Baboulin, J. Demmel, J. Dongarra, S. Tomov, V. Volkov, Enhancing the performance of dense linear algebra on GPUs. Supercomputing (SC 08), Austin, USA, Nov , [5] M. Baboulin, J. Dongarra, S. Tomov, Some issues in dense linear algebra for multicore and special purpose architectures. Springer LCNS Series, 9th International Workshop on State-of-the-Art in Scientific and Parallel Computing (PARA 08). Marc Baboulin 34/34 Accelerating linear algebra computations
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