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并行科学计算

作译者：弗雷德里克·马古雷斯（Frédéric Magoulès）,弗朗索瓦-泽维尔·鲁（Fran?ois-Xavier Roux）,邹秦萌

出版时间：2026-03 千字数：254 版次：01-01 页数：212

开本：16开装帧： I S B N ：9787121524462

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纸质书定价：￥98.0

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内容简介

前言

上架建议

作者简介

编辑推荐

音视频专区

第 1 章 计算机体系结构 ······························································.1
1.1 不同类型的并行 ·······························································.1
1.1.1 重叠、并发和并行 ·····················································.1 
1.1.2 算术逻辑单元中的空间并行与时间并行 ··························.3 
1.1.3 并行与内存······························································.5 
1.2 存储设备体系结构 ····························································.5 
1.2.1 多体交错式内存 ························································.6 
1.2.2 多级存储·································································.7 
1.2.3 分布式存储·····························································.11 
1.3 混合体系结构 ··································································12 
1.3.1 GPU 加速器 ·····························································12 
1.3.2 混合计算机······························································13 
第 2 章 并行化与编程模型 ···························································15 
2.1 并行化 ···········································································15 
2.2 性能指标 ········································································17 
2.2.1 并行度····································································17 
2.2.2 负载平衡·································································18 
2.2.3 粒度·······································································19 
2.2.4 可扩展性·································································19 
2.3 数据并行 ········································································20 
2.3.1 循环·······································································20 
2.3.2 依赖性····································································21 
2.3.3 依赖性示例······························································22 
2.3.4 归约运算·································································24
2.3.5 嵌套循环·································································25 
2.3.6 OpenMP ··································································29 
2.4 案例分析：向量化 ····························································31 
2.4.1 向量计算机和向量化 ··················································31 
2.4.2 依赖性····································································32 
2.4.3 归约运算·································································33 
2.4.4 流水线运算······························································34 
2.5 消息传递 ········································································36 
2.5.1 消息传递编程 ···························································36 
2.5.2 并行环境管理 ···························································36 
2.5.3 点对点通信······························································37 
2.5.4 聚合通信·································································38 
第 3 章 并行算法概念 ·································································41 
3.1 并行循环算法 ··································································41 
3.1.1 归约算法的基本原理 ··················································41 
3.1.2 归约算法的开销和稳定性 ············································42 
3.1.3 循环归约·································································44 
3.2 矩阵乘法的数据分布 ·························································45 
3.2.1 行算法和列算法 ························································45 
3.2.2 分块算法·································································47 
3.2.3 分布式算法······························································50 
3.2.4 算法实现·································································51 
第 4 章 矩阵计算基础 ·································································55 
4.1 线性代数回顾 ··································································55 
4.1.1 向量空间、点积、正交投影 ·········································55 
4.1.2 线性映射与矩阵 ························································58 
4.2 矩阵性质 ········································································61 
4.2.1 特殊矩阵、特征值、特征向量 ······································61 
4.2.2 矩阵范数·································································63 
4.2.3 基变换····································································65 
4.2.4 矩阵的条件数 ···························································66
第 5 章 稀疏矩阵 ·······································································71 
5.1 稀疏矩阵的起源 ·······························································71 
5.2 稀疏矩阵的并行构建：共享存储···········································75 
5.3 稀疏矩阵的分块并行构建：分布式存储 ··································75 
第 6 章 用 LU 分解法求解线性方程组·············································78 
6.1 线性方程组求解 ·······························································78 
6.1.1 直接法····································································78 
6.1.2 迭代法····································································79 
6.2 LU 分解的原理 ·································································81 
6.3 高斯消元法 ·····································································84 
6.4 高斯-若尔当消元法 ···························································86 
6.5 对称矩阵的 Crout 分解和楚列斯基分解 ··································91 
第 7 章 稠密矩阵 LU 分解的并行化················································94 
7.1 分块分解 ········································································94 
7.2 在 MPI 编程环境中实现分块分解 ··········································98 
7.3 前向-后向替换法的并行化 ···············································.102 
第 8 章 稀疏矩阵的 LU 分解 ······················································.104 
8.1 分解后的矩阵结构 ·························································.104 
8.2 符号分解与重新编号 ······················································.107 
8.3 消去树 ········································································.110 
8.4 消去树与依赖性 ····························································.115 
8.5 嵌套剖分 ·····································································.116 
8.6 前向-后向替换 ······························································.120 
第 9 章 对称正定矩阵的 Krylov 法·············································.123 
9.1 Krylov 子空间基础 ·························································.123 
9.1.1 Krylov 子空间························································.123 
9.1.2 Arnoldi 基的构建····················································.125 
9.2 对称矩阵 Lanczos 基的构建 ·············································.127 
9.3 Lanczos 法 ··································································.128 
9.4 共轭梯度法 ··································································.132 
9.5 与梯度法的对比 ····························································.136
9.6 对称正定矩阵的预处理 ···················································.138 
第 10 章 任意矩阵的正交化方法 ················································.141 
10.1 GMRES 法 ·································································.141 
10.2 面向对称矩阵的 MINRES 法 ···········································.148 
10.3 ORTHODIR 法 ····························································.151 
10.4 非对称矩阵的预处理 ·····················································.152 
第 11 章 非对称矩阵的双正交化方法··········································.154 
11.1 非对称矩阵双正交基的构建 ············································.154 
11.2 非对称 Lanczos 法 ·······················································.158 
11.3 双共轭梯度法 ······························································.159 
11.4 准极小残量法 ······························································.163 
11.5 稳定双共轭梯度法 ························································.167 
第 12 章 Krylov 法的并行化 ·····················································.173 
12.1 稠密矩阵-向量乘法的并行化 ··········································.173 
12.2 稀疏矩阵-向量乘法：按顶点划分·····································.174 
12.3 稀疏矩阵-向量乘法：按元素划分·····································.176 
12.3.1 区域分解原理概述 ················································.176 
12.3.2 矩阵-向量乘法 ·····················································.177 
12.3.3 数据通信 ····························································.178 
12.4 点积的并行化······························································.179 
12.5 Krylov 法的并行化小结··················································.180 
第 13 章 并行预处理方法 ·························································.182 
13.1 不完全分解法······························································.182 
13.1.1 原理 ··································································.182 
13.1.2 并行化 ·······························································.184 
13.2 舒尔补法····································································.186 
13.2.1 局部最优预处理 ···················································.186 
13.2.2 舒尔补法的原理 ···················································.187 
13.2.3 舒尔补法的性质 ···················································.189 
13.3 代数多重网格······························································.191 
13.3.1 基于投影的预处理 ················································.191
13.3.2 粗糙网格的代数构建 ·············································.193 
13.3.3 代数多重网格法 ···················································.195 
13.4 加性施瓦茨预处理法 ·····················································.196 
13.4.1 重叠原理 ····························································.196 
13.4.2 乘性方法与加性方法 ·············································.198 
13.4.3 加性 Schwarz 预处理法···········································.198 
结语························································································.201

目前，科学计算在力学、电磁学、生物学、金融、航空航天等领域已经成为必不可少的工具，研究者借助电子计算机和高性能算法，可以对雪后屋顶的形变、音乐厅的声音效果、机翼附近的空气流动等实际问题进行仿真，无须实物模型参与。
  本书主要针对偏微分方程离散化后得到的大规模问题，借助实例讲解和展示最新的科学计算技术，重点讨论大规模线性方程组的求解方法，如直接法、迭代法、区域分解法等。本书主要针对包含分布式存储和大量处理器的计算环境，因此也会介绍计算机基础知识以及基于消息传递接口的消息传递编程技术，但不会过多涉及计算机组成原理。此外，书中还将展示科学计算中的经典并行算法，相关示例来源于研究者关于有限元法等方法的实践经验。从教学角度看，本书循序渐进地介绍了数学与计算机的相关概念，包括机器的架构特性、并行管理的相关问题、矩阵计算的基础知识等，但仅面向科学计算工具的使用者，不会从计算机视角进行深入解读。
  本书包括以下内容：第 1 章“计算机体系结构”；第 2 章“并行化与编程模型”；第 3 章“并行算法概念”；第 4 章“矩阵计算基础”；第 5 章“稀疏矩阵”；第 6 章“用 LU 分解法求解线性方程组”；第 7 章“稠密矩阵 LU 分解的并行化”；第 8 章“稀疏矩阵的 LU 分解”；第 9 章“对称正定矩阵的 Krylov 法”；第 10章“任意矩阵的正交化方法”；第 11 章“非对称矩阵的双正交化方法”；第 12章“Krylov 法的并行化”；第 13 章“并行预处理方法”。
  本书的出版得到国家自然科学基金（项目编号：12571397）的资助，并得到北京邮电大学数学科学学院、数学与信息网络教育部重点实验室（北京邮电大学）的大力支持和帮助；北京邮电大学的研究生刘豪重新绘制了本书初稿中的全部图片，在此一并表示感谢。由于作者水平有限，书中疏漏之处在所难免，敬请广大读者批评和指正。
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