面向大规模并行计算的Algebraic Geometry
Towards Massively Parallel Computations in Algebraic Geometry
摘要：
引入并行性并探索其使用仍然是计算机代数社区的一个基本挑战。另一方面，在高性能数值模拟中，遵循协调与计算分离原则的分布式计算透明环境多年来一直是成功的。在本文中，我们将探讨在计算机代数的背景下使用这一原则的潜力。更准确地说，我们结合了两个完善的系统：我们感兴趣的数学在计算机代数系统Singular中实现，其重点是多项式计算，而协调则留给工作流管理系统GPI-Space，它依赖于Petri网作为其数学建模语言，并且已经成功地用于协调学术代码的并行执行（自动并行化）以及用于诸如地震数据处理的应用领域中的商业软件。我们的努力的结果是一个大规模并行计算的框架在奇异的应用领域，特别是在交换代数和代数几何的一个重要步骤。作为这个框架的第一个测试案例，我们已经建模并实现了混合光滑测试代数簇，它结合了Hironaka著名的desingularization证明与经典的雅可比标准的想法。将我们的实现应用于两个源自当前代数几何研究的示例，其中一个无法通过其他方式处理，我们说明了我们框架内平滑度测试的行为，并研究了计算如何扩展到256个核心。
英文摘要：
Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language, and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework, and investigate how the computations scale up to 256 cores.
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