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A Study on Mixed Precision Iterative Refinement using Low Precision Krylov Methods
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| Title: | A Study on Mixed Precision Iterative Refinement using Low Precision Krylov Methods |
| Other Titles: | 低精度クリロフ部分空間法を用いた混合精度反復改良法に関する研究 |
| Authors: | ZHAO, Yingqi Browse this author |
| Issue Date: | 25-Mar-2024 |
| Publisher: | Hokkaido University |
| Abstract: | Numerical linear algebra has wide applications in many aspects of life and is at the heart of important computational aspects of practical application problems. Numerical methods, often used to solve linear algebra problems, e.g., linear systems, eigenvalue problems, singular value decompositions, etc., in engineering and information ciences, play a key role and attract much attention. Among them, linear solvers for solving linear systems Ax = b are of vital importance in scientific computation and their computational time usually attracts much attention. An efficient linear solver can help accelerate practical applications. Therefore, the optimization of numerical methods and linear solvers is essential. Double-precision floating-point (also called FP64) has been regarded as a standard in most scientific computation for a long time. However, due to the reason of power budgets, the performance of FP64 has become more and more challenging and difficult to improve. Within recent years, some newly appeared hardware shows potentially high ability of low precision computing, some of which are equipped with specified units that can provide great performance in low precision computing, and have already been used in practical applications that do not strictly require high precision computing and accept low precision computing, i.e., machine learning. These circumstances promote the development of low precision computing while the output accuracy needs to be guaranteed. To handle this issue, one important strategy is to develop mixed precision (MP) algorithms that efficiently combine different precisions and achieve the same accuracy as the traditional method using only FP64. In this study, I focus on linear solvers and develop mixed precision algorithms for solving linear systems Ax = b whose coefficient matrix A is large, sparse, and non-symmetric. For this problem, the Krylov subspace methods are widely used in the field of numerical linear algebra. Some studies on mixed precision
computing for the Krylov subspace methods have already been reported. However, there are still many issues that need to be investigated. This study aims to provide new insights into mixed precision computing using the Krylov subspace methods, which will contribute to improving the computation performance of various applications that need liner solvers. In this study, two typical Krylov subspace methods, which are commonly used to solve non-symmetric linear systems, are considered. One is GMRES(m) method, and the other is BiCGSTAB method. Based on these two algorithms, I develop two mixed precision algorithms and conduct comprehensive numerical experiments using 26 test matrices selected from the SuiteSparse Matrix Collection. Through numerical experiments, I investigate the numerical characteristics and evaluate the effectiveness of the developed mixed precision algorithms in detail. In Chapter 3, a mixed precision variant of the GMRES(m) method using FP64 and FP32, which is called MP-GMRES(m), is investigated. Using the number of the inner iterations as restart frequency m, I have studied its numerical behavior through comprehensive experiments with different m from both theoretical and practical aspects. A detailed comparison with the traditional GMRES(m) method using only FP64 is also made. Detailed analysis and comparison of the obtained results are given from the following three aspects: the maximum attainable accuracy, the number of iterations, and the execution time. From the obtained results, MP-GMRES(m) has almost the same problem-solving ability as GMRES(m), and there is almost no difference in the final attainable accuracy if both two algorithms can not solve the problems. Although MP-GMRES(m) requires more number of iterations than GMRES(m), it provides a shorter execution time for most cases. I have also found some differences between these two methods; for example, as m increases, the number of iterations tends to
decrease in GMRES(m) while increasing in MP-GMRES(m). In Chapter 4, a mixed precision variant of iterative refinement using BiCGSTAB algorithm (MP-IR using BiCGSTAB) is developed and investigated in detail. In this algorithm, low precision(FP32) BiCGSTAB is employed as an inner solver in mixed precision iterative refinement, and two approaches are used for determining the restart: the number of the inner iterations m and the decrease of the residual 2-norm ϵ in the inner BiCGSTAB loop. Several sets of experiments are conducted, and the obtained results are analyzed and compared with the traditional BiCGSTAB method, as well as the MP-GMRES(m) method, which is studied in Chapter 3. The experiment results show that MP-IR using BiCGSTAB sometimes outperforms MP-GMRES(m) and is the fastest among all methods, especially for problems with small condition numbers, although MP-IR using BiCGSTAB is sensitive to a target problem. Similar to Chapter 3, comprehensive experimental results on the maximum attainable accuracy, the number of iterations, and the execution time are also provided and analyzed in this chapter. The rest of the thesis is organized as follows. In Chapter 1, I introduce the research background, purposes, and main contributions. The structure of the thesis is also briefly described. Chapter 2 summarizes the related work including mixed precision computing, the Krylov subspace methods, and some recent research on applying mixed precision computing to the Krylov subspace methods. Chapter 5 provides an overall summary of my research and discusses future work. In summary, the thesis aims to investigate the numerical characteristics and effectiveness of the mixed precision algorithms and show the applicability of mixed precision computing in numerical linear solvers. This study will provide options to accelerate the applications that need linear solvers and give some new insights into using the high ability of low precision computing, which hopes to promote further development of mixed precision linear solvers. |
| Conffering University: | 北海道大学 |
| Degree Report Number: | 甲第15998号 |
| Degree Level: | 博士 |
| Degree Discipline: | 情報科学 |
| Examination Committee Members: | (主査) 准教授 深谷 猛, 教授 棟朝 雅晴, 教授 飯田 勝吉, 客員教授 岩下 武史(北海道大学情報基盤センター) |
| Degree Affiliation: | 情報科学院(情報科学専攻) |
| Type: | theses (doctoral) |
| URI: | http://hdl.handle.net/2115/91870 |
| Appears in Collections: | 課程博士 (Doctorate by way of Advanced Course) > 情報科学院(Graduate School of Information Science and Technology)
学位論文 (Theses) > 博士 （情報科学）
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