有限元内核中的运行误差界
Running error bounds in finite element kernels
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中文总结 AI 辅助
本文提出首个基于运行误差分析的有限元内核自动化舍入误差估计框架，并开源实现，可检测灾难性抵消，性能开销仅2-4倍，适用于低精度计算与PDE调试。
中文摘要 AI 辅助
有限元计算中的舍入误差可能导致精度的完全丧失、收敛停滞以及错误的结果。此外，在自动生成并编译的内核中累积的舍入误差效应难以先验分析。我们提出了第一个用于有限元内核内自动化舍入误差估计的软件框架。所提出的方法基于一种称为运行误差分析（REA）的后验技术，其中前向误差估计与数值计算同时自动进行。我们为FEniCS表单编译器（FFCx）提供了开源实现，基于一个C++后端，用于生成类型通用的模板化内核，该内核基于一个自定义算术类型，同时跟踪数值及其误差估计。我们在两个示例上演示了REA。首先，我们使用它来检测小变形范围内Neo-Hooke超弹性模型组装中的灾难性抵消。通过级数展开规避了抵消问题，计算得到的误差估计证明了这一点。其次，我们研究了在近退化网格上拉普拉斯算子的组装。我们证明，我们的REA实现通常仅产生2-4倍的性能开销。这项工作的应用包括嵌入式系统中稳健的低精度计算、对新的可能病态或不稳定的PDE公式进行数值调试，以及指导混合精度内核的设计。
英文摘要
Rounding errors in finite element computations can lead to a complete loss of accuracy, stalled convergence, and incorrect results. Moreover, the effects of rounding errors accumulated within automatically generated and compiled kernels are difficult to analyze a priori. We present the first software framework for automated rounding error estimation within finite element kernels. The proposed methodology is based on an a posteriori technique called Running Error Analysis (REA), where a forward error estimate is automatically computed concurrently with the value. An open-source implementation is provided for the FEniCS Form Compiler (FFCx), based on a C++ backend for generating type-generic templated kernels over a custom arithmetic type that tracks both the value and its error estimate. We demonstrate REA on two examples. First, we use it to detect catastrophic cancellation in the assembly of a Neo-Hooke hyperelastic model in the small deformation regime. A series expansion circumvents the cancellation problem and the computed error estimates show this. Second, we study the assembly of the Laplace operator on a near-degenerate mesh. We demonstrate that our REA implementation typically incurs only 2-4x performance overhead. Applications of this work include robust reduced-precision computations in embedded systems, numerical debugging of new, possibly ill-conditioned or unstable PDE formulations, and guiding the design of mixed-precision kernels.
发表机构
- University of Luxembourg（卢森堡大学）
- Basque Center for Applied Mathematics（巴斯克应用数学中心）
- Ikerbasque（伊克拉巴斯科基金会）
- University of the Basque Country（巴斯克大学）
机构由 AI 辅助整理，请以论文原文为准。
