从低级退缩到动态低级近似和回归
Axel Séguin1, Gianluca Ceruti2, Daniel Kressner1
1École Polytechnique Fédérale de Lausanne (EPFL) Institute of Mathematics, 1015 Lausanne, Switzerland.
BIT. Numerical mathematics
|June 20, 2024
概括
收缩,用于保持在变形体上的计算的工具,应用于固定排列矩阵变形体上的微分方程的数值集成. 新的方法,加速前进欧勒 (AFE) 和预测拉斯顿 - 赫尔密特 (PRH),为动态低级近似 (DLRA) 提供第三阶精度.
科学领域:
- 数字分析 数字分析
- 微分几何学的差异几何学
- 矩阵计算 矩阵计算
背景情况:
- 收缩对于算法在光滑的多元体上解决优化问题至关重要.
- 收缩越来越多地用于基于多元体的计算任务,如插值.
- 在固定排列矩阵多元体上的微分方程的数值集成对于动态低排列近似 (DLRA) 至关重要.
研究的目的:
- 探索回收的应用在数值集成的微分方程在固定级别矩阵变量体.
- 引入基于撤回的新型数值整合方案.
- 在DLRA的背景下分析这些新方案的特性和性能.
主要方法:
- 使用收缩作为数字集成在多元组的核心组件.
- 介绍了从DLRA的非传统集成器中获得的KLS收缩.
- 开发两个新的整合方案:加速前进欧勒 (AFE) 和预计拉尔斯顿-赫米特 (PRH).
- 证明AFE和PRH方法的第三级局部截断误差.
主要成果:
- 证明了收缩自然会导致多重问题的数值集成器.
- 引入了KLS收缩作为DLRA的一个新工具.
- 展示了撤回如何可以统一现有的DLRA技术,并激发新的灵感.
- 对于新的AFE和PRH方法,确定的第三级局部截断错误.
结论:
- 收缩是适用于数值集成的多功能工具,适用于固定排列矩阵分组,与DLRA密切相关.
- 拟议的AFE和PRH方法提供了准确的 (第三阶) 数值积分方案,用于一般多元组的微分方程.
- 数值实验证实了DLRA的这些新的基于收缩的集成方法的实用性,并突出了这些新的基于收缩的集成方法的权衡.
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