Efficient Solver Algorithms for Graphical Processing Units

图形处理单元（gpu）在计算机仿真中的广泛应用正在改变大规模仿真。随着算法以峰值性能的一小部分运行，最近只有大型跨国公司和国家实验室才能实现的模拟现在甚至可以用于小型初创公司。不幸的是，编写能够利用gpu的峰值性能的算法并不容易。该项目专注于线性求解器，它通常占仿真时间的很大一部分，其目标是提高基于gpu的架构的效率。更具体地说，该项目旨在结合一系列技术来减少内存使用和数据移动（这在gpu上很耗时），更好的初始猜测和更好的停止标准（以适当的精度提前停止中间计算），以及动态更新求解器参数以提高效率。本项目的应用目标主要是计算流体动力学，但也包括逆问题和大规模拓扑优化。后者在新型微结构/元材料的开发、优化结构设计和制造中发挥着重要作用。该项目将包括一名研究生研究助理和一名博士后。该项目还将开发一门新的研究生课程，该课程结合了数值线性代数、数值常微分方程和偏微分方程以及GPU计算的元素，所有这些都聚焦于高性能。该项目涉及开发适合基于图形处理单元（GPU）架构的大规模问题的迭代求解器。这项工作旨在彻底重新评估求解器架构，以确保每个部分都针对GPU进行优化，在每个组件中暴露大量细粒度并行性，最大限度地提高GPU内存层次结构的每个级别的吞吐量，并最大限度地减少数据移动。该项目侧重于结合和分析三种关键策略的算法开发：混合精度变量和不精确矩阵向量积和平滑；计算更好的初始猜测和更有效的停止标准和指标；以及动态求解器优化和柔性预处理策略。结果将在能源部高效百亿亿次离散化中心开发的基准测试和该项目期间开发的类似基准测试中进行测试。本项目的理论基础将允许这些求解器策略产生超出本项目直接目标的实质性影响。为这个项目开发的开源算法和软件将免费提供给广大潜在用户。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The extensive use of graphical processing units (GPUs) for computer simulations is transforming large-scale simulations. With algorithms that run at a good fraction of peak performance, simulations that recently were only within reach for large multinational companies and national labs are now available even to small start-up companies. Unfortunately, it is not easy to write algorithms that can exploit the peak performance of GPUs. This project focuses on linear solvers, which generally account for a very large fraction of simulation time, with the goal of improving efficiency on GPU-based architectures. More specifically, the project aims to combine a range of techniques to reduce memory usage and data movement (which is time-consuming on GPUs), better initial guesses and better stopping criteria (to halt intermediate computations earlier at an appropriate precision), and dynamic updates to solver parameters to improve efficiency. Applications targeted in this project are primarily in computational fluid dynamics, but also include inverse problems and large-scale topology optimization. The latter plays a fundamental role in the development of new micro-structure/meta materials, their use in the design of optimal structures, and their manufacturing. The project will involve a graduate research assistant and a postdoc. The project will also develop a new graduate course that combines elements of numerical linear algebra, numerical ordinary and partial differential equations, and GPU computing, all with a focus on high performance.This project involves developing iterative solvers for large-scale problems suitable for graphics processing unit (GPU) based architectures. The work aims to thoroughly reevaluate solver architecture to ensure that every part is optimized for GPUs, exposing massive fine grain parallelism in every component, maximizing throughput at every level of the GPU memory hierarchy, and minimizing data movement. This project focuses on algorithmic development that combines and analyzes three key strategies: mixed precision variants and inexact matrix-vector products and smoothers; computing better initial guesses and more effective stopping criteria and indicators; and dynamic solver optimization and flexible preconditioning strategies. The results will be tested on benchmarks developed by the Department of Energy's Center for Efficient Exascale Discretizations and similar benchmarks to be developed during this project. The theoretical underpinnings from this project will allow these solver strategies to have substantial impact beyond the immediate goals of this project. The open-source algorithms and software developed for this project will be made freely available to a wide group of potential users.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.