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

Abstract

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.

Separated response surfaces for flows in parametrised domains: Comparison of a priori and a posteriori PGD algorithms

“…Indeed, a posteriori approaches feature an extended literature tackling various aspects critical for their efficiency, such as sampling strategies and error control. On the contrary, in the context of a priori ROMs, methodologies for the advanced treatment of the space of parameters [43] or the embedded control of accuracy [44][45][46] represent an active line of investigation. Hence, in order for the present comparison to be unbiased, similar versions of the a priori and a posteriori ROM algorithms, without targeted sampling or error control streatgies, are considered.…”

More Equilibrium, More Complementarity!

“…The orthogonality of the error of complementary solutions [2], was [3] and should continue to be, the main argument to nurture the development of equilibrium formulations. In recent years this has been reinforced with new ways to obtain local outputs, with smaller errors [4,5], that can be directly applied to reduced order models [6,7].…”

Static and dynamic global stiffness analysis for automotive pre-design

“…The PGD method was successfully tested in the most diverse classes of problems, such as flow problems [13][14][15][16][17][18], thermal problems [19][20][21], solid mechanics [22][23][24], fracture mechanics [25,26], geophysical problems [27,28], elastic metamaterials and coupled magneto-mechanical problems [29,30]. However, it still presents some limitations.…”