When designing a structure or engineering a component, it is crucial to use methods that provide fast and reliable solutions, so that a large number of design options can be assessed. In this context, the proper generalized decomposition (PGD) can be a powerful tool, as it provides solutions to parametric problems, without being affected by the “curse of dimensionality.” Assessing the accuracy of the solutions obtained with the PGD is still a relevant challenge, particularly when seeking quantities of interest with guaranteed bounds. In this work, we compute compatible and equilibrated PGD solutions and use them in a dual analysis to obtain quantities of interest and their bounds, which are guaranteed. We also use these complementary solutions to compute an error indicator, which is used to drive a mesh adaptivity process, oriented for a quantity of interest. The corresponding solutions have errors that are much lower than those obtained using a uniform refinement or an indicator based on the global error, as the proposed approach focuses on minimizing the error in the quantity of interest.

中文翻译：

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适当的广义分解解的误差估计：感兴趣的量的双重分析和适应性

在设计结构或工程部件时，至关重要的是使用提供快速和可靠解决方案的方法，以便可以评估大量的设计方案。在这种情况下，适当的广义分解（PGD）可以成为强大的工具，因为它可以为参数问题提供解决方案，而不受“维数的诅咒”的影响。评估使用PGD获得的解决方案的准确性仍然是一个挑战，尤其是在寻找具有保证范围的感兴趣数量时。在这项工作中，我们计算兼容且平衡的PGD解决方案，并在双重分析中使用它们，以确保有兴趣的数量及其范围。我们还使用这些补充解决方案来计算错误指标，该指标用于驱动网格自适应过程，面向大量利益。相应的解决方案所具有的误差远低于使用统一改进或基于全局误差的指标所获得的误差，这是因为所提出的方法着重于最大程度地减少关注数量的误差。