Solving Challenging Numerical Linear Algebra Algorithms Using GPU Accelerators Hatem Ltaief KAUST Supercomputing Laboratory Stanimire Tomov University of Tennessee, Knoxville ISC’12 Tutorial, Hamburg June 17, 2012

Outline MAGMA: LAPACK for GPUs Methodology overview Use both GPUs and multicore CPUs MAGMA: from single to multiGPU support One-sided factorizations and linear solvers Two-sided factorizations and eigensolvers Dynamic scheduling approaches to DLA MAGMA algorithms with dynamic scheduling Conclusions

MAGMA: LAPACK for GPUs MAGMA Matrix algebra for GPU and multicore architecture To provide LAPACK/ScaLAPACK on hybrid architectures MAGMA BLAS A subset of BLAS for GPUs, highly optimized for NVIDIA GPGPUs Fast GEMM for Fermi (CUBLAS3.2) [IJHPCA’10] MAGMA developers & collaborators UTK, UC Berkeley, UC Denver, INRIA (France), KAUST (Saudi Arabia) Community effort, similarly to LAPACK/ScaLAPACK

A New Generation of DLA Software MAGMA Hybrid Algorithms (heterogeneity friendly) Rely on - hybrid scheduler - hybrid kernels

MAGMA Software Stack

MAGMA 1.1  50+ hybrid LAPACK algorithms have been developed (total of 200+ routines)  Every algorithm is in 4 precisions (s/c/d/z)  There are 3 mixed precision algorithms (zc & ds)  These are hybrid algorithms, expressed in terms of BLAS  MAGMA BLAS  A subset of GPU BLAS, optimized for Tesla and Fermi GPUs

MAGMA Methodology A methodology to use all available resources:  MAGMA uses HYBRIDIZATION methodology based on – Representing linear algebra algorithms as collections of TASKS and DATA DEPENDENCIES among them – Properly SCHEDULING tasks' execution over multicore and GPU hardware components  Successfully applied to fundamental linear algebra algorithms – One and two-sided factorizations and solvers – Iterative linear and eigen-solvers  Productivity – 1) High-level; 2) Leveraging prior developments; 3) Exceeding in performance homogeneous solutions Hybrid CPU+GPU algorithms (small tasks for multicores and large tasks for GPUs)

Hybrid Algorithms One-sided factorizations (LU, QR, Cholesky) Hybridization – Panels (Level 2 BLAS) are factored on CPU using LAPACK – Trailing matrix updates (Level 3 BLAS) are done on the GPU using “look-ahead”

A Hybrid Algorithm Example  Left-looking hybrid Cholesky factorization in MAGMA 1.0  The difference with LAPACK – the 3 additional lines in red  Line 10 (done on CPU) is overlapped with work on the GPU

LU Factorization (Single GPU)

From single to multiple GPUs support Data distribution 1-D block-cyclic distribution Algorithm GPU holding current panel is sending it to CPU All updates are done in parallel on the GPUs Look-ahead is done with GPU holding the next panel GPU 0 GPU 1 GPU 2 GPU 0... nb

LU factorization (multiGPUs)

Matrix out of GPU memory

Out of GPU Memory Algorithms Perform left-looking factorizations on sub-matrices that fit in the GPU memory (using existing algorithms) The rest of the matrix stays on the CPU Left-looking versions minimize writing on the CPU Factored sub-matric A 1 on CPU To be factored sub- matrix A 2 on GPU... 1)Copy A 2 to the GPU 2)Update A 2 using A 1 (a panel of A 1 at a time) 3)Factor the updated A 2 using existing hybrid code 4)Copy factored A 2 to the CPU Trivially extended to multiGPUs: A 2 is “larger” with 1-D block cyclic distribution, again reusing existing algorithms Untouched part of the matrix

Hybrid Algorithms Two-sided factorizations (to bidiagonal, tridiagonal, and upper Hessenberg forms) for eigen- and singular-value problems Hybridization – Trailing matrix updates (Level 3 BLAS) are done on the GPU (similar to the one-sided factorizations) – Panels (Level 2 BLAS) are hybrid – operations with memory footprint restricted to the panel are done on CPU – The time consuming matrix-vector products involving the entire trailing matrix are done on the GPU

MultiGPUs Two-Sided Factorizations Performance of DSYMV on multi M2090s Need HP multiGPU Level 2 BLAS (e.g., 50% of flops in the tridiagonal reduction) T. Dong, J. Dongarra, S. Tomov, I. Yamazaki, T. Schulthess, and R. Solca, Symmetric dense matrix-vector multiplication on multiple GPUs and its application to symmetric dense and sparse eigenvalue problems, ICL Technical report, 03/2012.

Hybrid Two-Sided Factorizations

MAGMA Tridiagonalization in DP T. Dong, J. Dongarra, S. Tomov, I. Yamazaki, T. Schulthess, and R. Solca, Symmetric dense matrix-vector multiplication on multiple GPUs and its application to symmetric dense and sparse eigenvalue problems, ICL Technical report, 03/2012.  50 % of the flops are in SYMV  Memory bound, i.e. does not scale well on multicore CPUs  Use the GPU’s high memory bandwidth and optimized SYMV  8 x speedup over 12 Intel cores GHz) Keeneland system, using one node 3 NVIDIA GPUs 1.1 GHz, 5.4 GB) 2 x 6 Intel Cores 2.8 GHz, 23 GB)

Further GPU Kernel Optimizations Fermi 2070 A. Abdelfattah, J. Dongarra, D. Keyes and H. Ltaief, Optimizing Memory-Bound Numerical Kernels on GPU Hardware Accelerators, VECPAR, Japan, 2012.

Further GPU Kernel Optimizations Fermi 2070

Further GPU Kernel Optimizations Fermi 2070

From Static to Dynamic Scheduling… Static may stall in situations where work is available Hand tuned optimizations Hardware heterogeneity Kernel heterogeneity Separation of concerns Dynamic Runtime System

Punch Lines Productivity! Oh… Did I say Productivity?

Block Algorithms Panel-Update Sequence Transformations are blocked/accumulated within the Panel (Level 2 BLAS) Transformations applied at once on the trailing submatrix (Level 3 BLAS) Parallelism hidden inside the BLAS Fork-join Model

Block QR Factorization

Fork-Join Paradigm

Leveraging Block Algorithms… Column-major data layout Tile data layout

Lessons Learnt from PLASMA PLASMA: Parallel Linear Algebra for Scalable Multi-core Architectures: Tile Algorithms on homogeneous x86 cores Parallelism is brought to the fore May require the redesign of linear algebra algorithms Tile data layout translation Remove unnecessary synchronization points between Panel- Update sequences DAG execution where nodes represent tasks and edges define dependencies between them Dynamic runtime system environment QUARK

Example: Tile QR Factorization First panel factorization and corresponding updates DAG for a 4x4 tiles matrix

Let’s go crazy! DAG of 20x20 tile QR Factorization

Dynamic Scheduling Conceptually similar to out-of-order processor scheduling because it has: Dynamic runtime DAG scheduler Out-of-order execution flow of fine-grained tasks Task scheduling as soon as dependencies are satisfied Producer-Consumer Data Flow Programming Model: five decades old concept Think "how things connect" rather than "how things happen” Assembly line Inherently parallel

Matrices Over Runtime Systems at Exascale (MORSE) Mission statement: "Design dense and sparse linear algebra methods that achieve the fastest possible time to an accurate solution on large-scale Hybrid systems” Runtime challenges due to the ever growing hardware complexity Algorithmic challenges to exploit the hardware capabilities at most Integrated into MAGMA software stack

MAGMA-MORSE: x86 + MultiGPUs Lessons Learned from PLASMA! CUDA-based hybrid systems New high performance numerical kernels StarPU Runtime System (Augonnet et. Al, INRIA, Bordeaux) Both: x86 and GPUs => Hybrid Computations Similar to LAPACK in functionality

Achieving High Level of Productivity From Sequential Nested-Loop Code to Parallel Execution for (k = 0; k < min(MT, NT); k++){ zgeqrt(A[k;k],...); for (n = k+1; n < NT; n++) zunmqr(A[k;k], A[k;n],...); for (m = k+1; m < MT; m++){ ztsqrt(A[k;k],,A[m;k],...); for (n = k+1; n < NT; n++) ztsmqr(A[m;k], A[k;n], A[m;n],...); }

Achieving High Level of Productivity From Sequential Nested-Loop Code to Parallel Execution for (k = 0; k < min(MT, NT); k++){ starpu_Insert_Task(&cl_zgeqrt, k, k,...); for (n = k+1; n < NT; n++) starpu_Insert_Task(&cl_zunmqr, k, n,...); for (m = k+1; m < MT; m++){ starpu_Insert_Task(&cl_ztsqrt, m, k,...); for (n = k+1; n < NT; n++) starpu_Insert_Task(&cl_ztsmqr, m, n, k,...); }

Hybrid Architecture Targeted ⇒ PCI Interconnect 16X 64Gb/s, very thin pipe! ⇒ Fermi C cuda cores 515 Gflop/s

Performance Charts Cholesky QR LU Symmetric Matrix Inversion

Cholesky Performance 8 Intel x86 cores + 3 Tesla GPUs E. Agullo, C. Augonnet, J. Dongarra, H. Ltaief, S. Thibault, R. Namyst, S. Thibault, S. Tomov, Software for GPUs, GPU Computing GEMs, vol.2, 2011.

QR Performance E. Agullo, C. Augonnet, J. Dongarra, M. Faverge, H. Ltaief, S. Thibault, S. Tomov, IEEE International Parallel and Distributed Processing Symposium, AMD x86 cores + 4 Tesla GPUs

QR Performance +~200 Gflop/s but 12 cores = ~150 Gflop/s 8 AMD x86 cores + 4 Tesla GPUs E. Agullo, C. Augonnet, J. Dongarra, M. Faverge, H. Ltaief, S. Thibault, S. Tomov, IEEE International Parallel and Distributed Processing Symposium, 2011.

Performance Breakdown Task distribution observed on StarPU: sgeqrt : 20% of tasks on GPUs stsmqr : 92.5% of tasks on GPUs Taking advantage of heterogeneity ! Only do what you are good for Don’t do what you are not good for KernelCPUGPUSpeedup sgeqrt9 Gflops60 Gflops~ 6 stsqrt12 Gflops67 Gflops~ 6 sormqr8.5 Gflops227 Gflops~ 27 stsmqr10 Gflops285 Gflops~ 27

LU Performance E. Agullo, C. Augonnet, J. Dongarra, M. Faverge, J. Langou, H. Ltaief and S. Tomov, ACS/IEEE International Conference on Computer Systems and Applications (best paper award), Intel x86 cores + 3 Tesla GPUs

Symmetric Matrix Inversion A −1, Seriously??? YES! Critical component of the variance-covariance matrix computation in statistics Three steps: Cholesky factorization (DPOTRF) Inverting the Cholesky factor (DTRTRI) Calculating the product of the inverted Cholesky factor with its transpose (DLAUUM) Built on previous work from E. Agullo, H. Bouwmeester, J. Dongarra, J. Kurzak, J. Langou, and L. Rosenberg

Scheduling Algorithms as DAGs 44

A −1 Performance H. Ibeid, D. Kaushik, D. Keyes and H. Ltaief, HIPC'11, India 8 Intel x86 cores + 2 Fermi GPUs

Summary and Future Directions Two methodologies for solving challenging DLA Static Scheduling (performance) Dynamic Scheduling (productivity) LAPACK compliant API Source codes freely available in MAGMA What’s next? Extended numerical functionality Distributed Memory Systems

Colloborators / Support  MAGMA [Matrix Algebra on GPU and Multicore Architectures] team  PLASMA [Parallel Linear Algebra for Scalable Multicore Architectures] team  Collaborating partners University of Tennessee, Knoxville University of California, Berkeley University of Colorado, Denver INRIA, France KAUST, Saudi Arabia

Questions?