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Error estimation and adaptivity for PGD based on complementary solutions applied to a simple 1D problem
Advanced Modeling and Simulation in Engineering Sciences Pub Date : 2020-10-30 , DOI: 10.1186/s40323-020-00180-3
Jonatha Reis , José Paulo Moitinho de Almeida , Pedro Díez , Sergio Zlotnik

Reduced order methods are powerful tools for the design and analysis of sophisticated systems, reducing computational costs and speeding up the development process. Among these reduced order methods, the Proper Generalized Decomposition is a well-established one, commonly used to deal with multi-dimensional problems that often suffer from the curse of dimensionality. Although the PGD method has been around for some time now, it still lacks mechanisms to assess the quality of the solutions obtained. This paper explores the dual error analysis in the scope of the PGD, using complementary solutions to compute error bounds and drive an adaptivity process, applied to a simple 1D problem. The energy of the error obtained from the dual analysis is used to determine the quality of the PGD approximations. We define a new adaptivity indicator based on the energy of the error and use it to drive parametric h- and p- adaptivity processes. The results are positive, with the indicator accurately capturing the parameter that will lead to lowest errors.

中文翻译:

基于应用于简单一维问题的互补解的PGD的误差估计和适应性

降阶方法是用于设计和分析复杂系统,降低计算成本并加快开发过程的强大工具。在这些降阶方法中,“适当的广义分解”是一种行之有效的方法,通常用于处理经常遭受维度诅咒的多维问题。尽管PGD方法已经存在了一段时间,但它仍然缺乏评估获得的解决方案质量的机制。本文探讨了PGD范围内的双重误差分析,使用补充解决方案来计算误差范围并驱动适应性过程,并将其应用于简单的一维问题。从对偶分析获得的误差能量用于确定PGD近似值的质量。我们基于误差的能量定义了一个新的适应性指标,并使用它来驱动参数h和p适应过程。结果是肯定的,指示器准确地捕获了将导致最小错误的参数。
更新日期:2020-10-30
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