Conjugate Gradient Solvers with High Accuracy and Bit-wise Reproducibility between CPU and GPU using Ozaki Scheme
| Autor: | Takeshi Ogita, Daichi Mukunoki, Katsuhisa Ozaki, Roman Iakymchuk |
|---|---|
| Přispěvatelé: | RIKEN Center for Computational Science [Kobe] (RIKEN CCS), RIKEN - Institute of Physical and Chemical Research [Japon] (RIKEN), Shibaura Institute of Technology, Tokyo Woman's Christian University, Department of Mathematics, Tokyo Woman's Christian University (TWCU), Performance et Qualité des Algorithmes Numériques (PEQUAN), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Iakymchuk, Roman |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Computer science
Computation GPU 010103 numerical & computational mathematics 02 engineering and technology Parallel computing 01 natural sciences Basic Linear Algebra Subprograms Conjugate gradient method Convergence (routing) [INFO.INFO-DC] Computer Science [cs]/Distributed Parallel and Cluster Computing [cs.DC] 0202 electrical engineering electronic engineering information engineering Conjugate Gradient 0101 mathematics Bitwise operation Accuracy [INFO.INFO-MS]Computer Science [cs]/Mathematical Software [cs.MS] 020203 distributed computing Rounding [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic Krylov subspace [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA] Reproducibility Nondeterministic algorithm Heterogenous computing [INFO.INFO-MS] Computer Science [cs]/Mathematical Software [cs.MS] [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic CPU [INFO.INFO-DC]Computer Science [cs]/Distributed Parallel and Cluster Computing [cs.DC] [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| Zdroj: | HPC Asia The International Conference on High Performance Computing in Asia-Pacific Region |
| Popis: | On Krylov subspace methods such as the Conjugate Gradient (CG), the number of iterations until convergence may increase due to the loss of computation accuracy caused by rounding errors in floating-point computations. Besides, as the order of operations is non-deterministic on parallel computations, the result and the behavior of the convergence may be non-identical in different environments, even for the same input. This paper presents a new approach for the CG method with high accuracy as well as bit-level reproducibility of computed solutions on many-core processors, including both x86 CPUs and NVIDIA GPUs. In our proposed approach, accurate and reproducible operations are installed into all the inner-product based operations such as matrix-vector multiplication and dot-product, which are the main sources that may disturb reproducibility in the CG method. The accurate and reproducible operations are performed using the Ozaki scheme, which is the error-free transformation for dot-product that can ensure the correct-rounding. As this method can be built upon vendor-provided linear algebra libraries such as Intel Math Kernel Library and NVIDIA cuBLAS/ cuSparse, it reduces the development cost. In this paper, showing some examples with the non-identical conver-gences and computed solutions on different platforms, we demonstrate the applicability and the effectiveness of the proposed approach as well as its performance on both CPUs and GPUs. Besides, we compare against an existing accurate and reproducible CG implementation based on the Exact BLAS (ExBLAS) on CPUs. |
| Databáze: | OpenAIRE |
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