Error estimation for proper generalized decomposition solutions: Dual analysis and adaptivity for quantities of interest

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.

Automated summary

In this work, compatible and equilibrated PGD solutions are computed and used in a dual analysis to obtain quantities of interest and their bounds with guaranteed accuracy. Complementary solutions are also used to compute an error indicator, driving a mesh adaptivity process focused on minimizing errors in the quantity of interest. The proposed approach results in lower errors compared to uniform refinement or global error-based indicators.