1. Graphics Processing Units (GPU) General-Purpose GPU (GPGPU) GPU TSUBAME 2.0[1] GPU 515 GFlops 150 GB/s GPU [2], [3], [4], [5], [6], [7], [8] GPU NV
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1 AMR GPU 1,a) 1,b) 1,c) , Adaptive mesh refinement (AMR) GPU CPU GPU AMR GPU AMR GPU CPU GPU AMR AMR GPU C++ 3 GPU Framework for Block-type AMR method on GPU computing Shimokawabe Takashi 1,a) Aoki Takayuki 1,b) Onodera Naoyuki 1,c) Received: November 4, 2011, Accepted: December 1, 2011 Abstract: The paper proposes a framework for block-type adaptive mesh refinement (AMR) methd on GPU computing. In large-scale computation, AMR methods are used for computing locally at high resolution. However, many of previous works on mesh-based applications on GPUs exploited Cartesian grids due to GPU architecture. Our framework automatically translates user-written functions that update a grid point and generates both GPU and CPU code. The framework can hide optimizations for GPUs and complicated implementation for the AMR method. This framework allows us to apply the AMR method to various mesh-based computations easily. It is implemented using the C++ language in order for easily porting to other environments. In addition, in order to calculate high-precision numerical methods with high accuracy, the data transfer between the different resolutions is performed by using 3rd-order accuracy interpolation function. As an experiment evaluation, we have implemented advection calculations and phase-field method by using this framework and have achieved faster computation with high accuracy using the AMR method. Keywords: GPU, mesh-based applications, framework, adaptive mesh refinement 1 Tokyo Institute of Technology i7-3, Ohokayama, Meguro-ku, Tokyo, Japan a) shimokawabe@sim.gsic.titech.ac.jp b) taoki@gsic.titech.ac.jp c) onodera@sim.gsic.titech.ac.jp c 2013 Information Processing Society of Japan 156
2 1. Graphics Processing Units (GPU) General-Purpose GPU (GPGPU) GPU TSUBAME 2.0[1] GPU 515 GFlops 150 GB/s GPU [2], [3], [4], [5], [6], [7], [8] GPU NVIDIA GPU CUDA[9] OpenCL[10] Domain specific language; DSL Physis [11] CUDA GPU Mint [12] TSUBAME 2.0 GPU GPU [2], [4], [8] CPU GPU GPU GPU (Adaptive mesh refinement; AMR) [13], [14], [15] GPU AMR [16], [17] AMR GPU AMR Physis DSL GPU AMR GPU AMR C/C++ CUDA AMR AMR GPU GPU GPU AMR GPU CPU 2. AMR AMR AMR (1) (2) GPU (2) AMR NVIDIA GPU C/C++ CUDA GPU AMR GPU AMR CPU AMR C++ C++ AMR c 2013 Information Processing Society of Japan 157
3 (1) (2) (1) (2) AMR 3. AMR AMR AMR 3.1 AMR NVIDIA GPU 1warp thread GPU 3.2 AMR 3 1 AMR Fig. 1 A blocked AMR grid in 2D i f i+ 1 2 f i+ 1 2 = 1 16 (f i+2 9f i+1 9f i + f i 1 ) (1) 4 AMR 2 [16] AMR 2 AMR TVD Runge-Kutta 4 1 L AMR x AMR x 2 x x = x 0 2 x 0, 4 x 0, 8 x 0, 16 x 0, 32 x 0 AMR 3 4 AMR 4 AMR AMR 3 c 2013 Information Processing Society of Japan 158
4 2 Fig. 2 Error 3 3 Fig Data access required for 3rd-order polynomial interpolation. Data access required for linear interpolation. w/ AMR (linear) w/ AMR (3rd-order polynomial) w/o AMR x Fig Accuracy of 3rd-order upwind scheme (1) (Request) (2) (Data copy) 5 (1) 1 2 (2) (1) 1 2 (3) request 1 request 2 request 2 request 3 data copy (1) (2) (3) 3.4 AMR c 2013 Information Processing Society of Japan 159
5 (1) (2) (3) Request request request request 2 request 1 request 3 Data copy data copy data copy data copy 2 data copy 1 Fig. 5 5 Data copy required by the resolution change of. GPU AMR CUDA 1 block AMR CUDA 1 thread CUDA block (256, 1, 1) 1 AMR thread 1 AMR 1 CUDA block AMR GPU CUDA block threads [2], [3], [4] AMR CUDA block AMR GPU CUDA block AMR CUDA block AMR 4. C++ CUDA C++ AMR AMR (1) AMR (2) (3) AMR AMR AMR 4.1 AMR C++ AMRTables AMR AMR AMRTables AMR n f n n + 1 f n+1 2 AMRTables 2 AMRTables.swap(0, 2) 0 2 c 2013 Information Processing Society of Japan 160
6 AMRTables CPU GPU AMR AMRTables 4.2 C++ ArrayIndex3D 3 ArrayIndex3D (n x, n y, n z ) (i, j, k) f ArrayIndex3D.ix() f[arrayindex3d.ix()] f (i, j, k) ArrayIndex3D ArrayIndex3D.ix<+1, 0, 0>() ArrayIndex3D.ix<-1, -2, 0>() (i + 1, j, k) (i 1, j 2, k) GPU CPU 4.3 ArrayIndex3D run 2 struct StencilFunc { host device static void run ( const ArrayIndex2D &bidx, const ArrayIndex2D &idx, float **f, float **fn, float a0, float a1,...) { float b = a0*f[bidx.ix()][idx.ix<+1, 0>()] + a1*f[bidx.ix()][idx.ix< 0, +1>()];... } }; AMR bidx, idx (i, j, k) (i, j, k) f, fn AMR f, fn f, fn, f, fn AMRTables AMRTables amrtables amrtables.run<stencilfunc>(f, fn, a0, a1,...) AMRTables.run() C++ AMRTables.run() StencilFunc::run() StencilFunc::run() host device AMRTables CPU GPU AMRTables CPU StencilFunc::run() for GPU CUDA CUDA block thread 4.4 (i, j, k) ArrayIndex3D 2 template <typename T> struct DataCopy { host device static void copy_to_higherlevel( const ArrayIndex2D &bidx, const ArrayIndex2D &idx, const T **p, ArrayIndex2D *idxn, T *pn,...) {... c 2013 Information Processing Society of Japan 161
7 }; }... AMR bidx idx idxn p pn bidx idx (i, j, k) (i, j, k) AMRTables host device AMRTables CPU GPU AMR AMR AMR 1 φ(r) = 2 0 (r R) [ 1 + cos ( πr )] (r < R) R r (2) x y AMR AMR 5.2 [4], [18], [19] AMR GPU NVIDIA Tesla C2050 CUDA AMR 8 AMR 4 w/ AMR AMR 1 w/o AMR 1 Single mesh Single mesh w/o AMR 8 AMR 7 w/ AMR (Single mesh) AMR (w/ AMR) w/ AMR Single mesh 0.6 AMR 8 w/o AMR Single mesh Single mesh w/o AMR AMR c 2013 Information Processing Society of Japan 162
8 2013年ハイパフォーマンスコンピューティングと計算科学シンポジウム 図 6 AMR 法を用いたコサインベルの移流計算 Fig. 6 Advection of cosine bell profile with the AMR method. 図 7 AMR 法を用いたフェーズフィールド計算 Fig. 7 Phase-field simulation with the AMR method. AMR ブロックの配列へのポインタを配列として保持する 必要がある このため 実データへのアクセスがポインタ への配列を経由して行われる GPU のデバイスメモリは 遅く このポインタへの配列を用いた参照は性能向上のボ トルネックとなる また アドレス計算が比較的複雑であ り 分断されたメモリ領域へのアクセスが発生する これ らも性能低下の原因となる 一方で AMR 法を用いず計 算領域全体の物理変数を単一の配列に保持する場合は ア ドレス計算が比較的簡単であり また配列へのアクセス時 20 Computational time [msec] ものである 提案フレームワークを含め AMR 法では 15 w/ AMR w/o AMR Single mesh のキャッシュヒット率が比較的高くなることが期待でき る 計算領域全体を最高解像度で計算する際は データ構 造がより単純な AMR 法を用いず計算領域全体の物理変数 を単一の配列に保持した計算方法が有利となる 図 9 は 図 8 と同じ計算における フェーズフィール ド計算のあるタイムステップまでの計算にかかった累積時 間を示す 計算の初期では AMR 法 w/ AMR は最も 高速に目標タイムステップまで計算できている 計算が先 Timestep 図 8 フェーズフィールド計算の各タイムステップの実行時間 Fig. 8 Computational times taken by each timestep of phasefield simulations. は GPU を対象としているが CPU においても十分に高い 性能を達成できている 6. おわりに に進むにつれ フレームワークを使用しない計算 Single ブロック AMR 法の GPU コンピューティング フレー mesh が短時間で目標のタイムステップまで計算を行え ムワークを提案した いくつかの格子計算用のフレーム る ステップあたりを過ぎると累計計算時間にお ワークは 存在するものの GPU で実行する AMR 法を いても AMR 法は優位ではなくなる 対象としたフレームワークは知られていない 提案フレー 本フレームワークは GPU コード生成時と同時に CPU ムワークは ステンシル計算を簡便に記述するためのテン コードも生成する 計算開始直後では CPU で AMR 法 プレート関数とクラスを提供する プログラマは これら を用いることで 直交格子を単一配列として確保し計算し を用い格子点上で計算についてのみ記述することで 簡単 た場合の 16.2 倍の高速化を達成した 本フレームワーク に格子計算に AMR を導入することが可能である 提案フ 2013 Information Processing Society of Japan 163
9 Accumulated elapsed time [min] Fig w/ AMR w/o AMR Single mesh Timestep 9 Accumulated execution times of phase-field simulations. C/C++ CUDA GPU 3 AMR CUDA block AMR CUDA 1 thread AMR AMR AMR AMR 3 AMR 2 AMR AMR B GPU CREST [1] Endo, T., Nukada, A., Matsuoka, S. and Maruyama, N.: Linpack Evaluation on a Supercomputer with Heterogeneous Accelerators, Proceedings of the 24th IEEE International Parallel and Distributed Processing Symposium (IPDPS 10), Atlanta, GA, USA, IEEE (2010). [2] Shimokawabe, T., Aoki, T., Muroi, C., Ishida, J., Kawano, K., Endo, T., Nukada, A., Maruyama, N. and Matsuoka, S.: An 80-Fold Speedup, 15.0 TFlops Full GPU Acceleration of Non-Hydrostatic Weather Model ASUCA Production Code, Proceedings of the 2010 ACM/IEEE International Conference for Computing, Networking, Storage and Analysis, SC 10, New Orleans, LA, USA, IEEE Computer Society, pp (online), DOI: (2010). [3] Shimokawabe, T., Aoki, T., Ishida, J., Kawano, K. and Muroi, C.: 145 TFlops Performance on 3990 GPUs of TSUBAME 2.0 Supercomputer for an Operational Weather Prediction, Procedia Computer Science, Vol. 4, pp (online), DOI: DOI: /j.procs (2011). Proceedings of the International Conference on Computational Science, ICCS [4] Shimokawabe, T., Aoki, T., Takaki, T., Yamanaka, A., Nukada, A., Endo, T., Maruyama, N. and Matsuoka, S.: Peta-scale Phase-Field Simulation for Dendritic Solidification on the TSUBAME 2.0 Supercomputer, Proceedings of the 2011 ACM/IEEE International Conference for Computing, Networking, Storage and Analysis, SC 11, Seattle, WA, USA, ACM, pp (2011). [5] Michalakes, J. and Vachharajani, M.: GPU acceleration of numerical weather prediction., IPDPS, IEEE, pp. 1 7 (2008). [6] Linford, J. C., Michalakes, J., Vachharajani, M. and Sandu, A.: Multi-core acceleration of chemical kinetics for simulation and prediction, SC 09: Proceedings of the Conference on Computing Networking, Storage and Analysis, New York, NY, USA, ACM, pp (online), DOI: (2009). [7] Hamada, T. and Nitadori, K.: 190 TFlops Astrophysical N-body Simulation on a Cluster of GPUs, Proceedings of the 2010 ACM/IEEE International Conference for Computing, Networking, Storage and Analysis, SC 10, New Orleans, LA, USA, IEEE Computer Society, pp. 1 9 (online), DOI: (2010). [8] Feichtinger, C., Habich, J., Köstler, H., Hager, G., Rüde, c 2013 Information Processing Society of Japan 164
10 U. and Wellein, G.: A Flexible Patch-Based Lattice Boltzmann Parallelization Approach for Heterogeneous GPU CPU Clusters, Parallel Computing, Vol. 37, No. 9, pp (2011). [9] NVIDIA: CUDA C Programming Guide 4.0, http: //developer.download.nvidia.com/compute/cuda/ 4 0/toolkit/docs/CUDA C Programming Guide.pdf (2011). [10] Khronos OpenCL Working Group: The OpenCL Specification, version (2008). [11] Maruyama, N., Nomura, T., Sato, K. and Matsuoka, S.: Physis: an implicitly parallel programming model for stencil computations on large-scale GPUaccelerated supercomputers, Proceedings of 2011 International Conference for Computing, Networking, Storage and Analysis, SC 11, New York, NY, USA, ACM, pp. 11:1 11:12 (online), DOI: (2011). [12] Unat, D., Cai, X. and Baden, S. B.: Mint: realizing CUDA performance in 3D stencil methods with annotated C, Proceedings of the international conference on Supercomputing, ICS 11, New York, NY, USA, ACM, pp (online), DOI: (2011). [13] Berger, M. J. and Oliger, J.: Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, Journal of Computational Physics, Vol. 53, p. 484 (online), DOI: / (84) (1984). [14] Berger, M. J. and Colella, P.: Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, Vol. 82, pp (online), DOI: / (89) (1989). [15] Fryxell, B., Olson, K., Ricker, P., Timmes, F. X., Zingale, M., Lamb, D. Q., MacNeice, P., Rosner, R., Truran, J. W. and Tufo, H.: FLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes, The Astrophysical Journal Supplement Series, Vol. 131, No. 1, p. 273 (online), available from (2000). [16] Schive, H.-Y., Tsai, Y.-C. and Chiueh, T.: GAMER: A Graphic Processing Unit Accelerated Adaptive-Mesh- Refinement Code for Astrophysics, The Astrophysical Journal Supplement Series, Vol. 186, No. 2, p. 457 (online), available from (2010). [17] Shukla, H., Schive, H.-Y., Woo, T.-P. and Chiueh, T.: Multi-science applications with single codebase - GAMER - for massively parallel architectures, Proceedings of 2011 International Conference for Computing, Networking, Storage and Analysis, SC 11, New York, NY, USA, ACM, pp. 37:1 37:11 (online), DOI: / (2011). [18] Tiaden, J., Nestler, B., Diepers, H. J. and Steinbach, I.: The multiphase-field model with an integrated concept for modelling solute diffusion, Physica D: Nonlinear Phenomena, Vol. 115, No. 1-2, pp (online), DOI: DOI: /S (97) (1998). [19] Kim, S. G., Kim, W. T. and Suzuki, T.: Phasefield model for binary alloys, Phys. Rev. E, Vol. 60, No. 6, pp (online), DOI: /Phys- RevE (1999). c 2013 Information Processing Society of Japan 165
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