A posteriori verification of PGD reduced models with application to engineering computation
Résumé
Despite the important progress in computer sciences, the cost associated with the resolution of multi- parametric problems can be extremely prohibitive. This is especially important when dealing with problems depending on numerous parameters, as encountered in optimization studies, which are becoming more and more mandatory for the design of new products. Reduced Order Modelling (ROM) is a convenient answer to circumvent this issue, usually called curse of dimensionality. However, a main drawback of ROM is the lack of robust a posteriori error estimators to measure the quality of the approximated solution, even if first advances have been performed [1,2].
This paper extends the a posteriori verification procedure proposed in [3] that enables to control and certify PGD-based model reduction techniques, which are currently the subject of many research activities [4]. This procedure uses the concept of constitutive relation error allowing to get guaranteed global/goal-oriented error estimator taking both discretization and PGD truncation error into account. By splitting the errors sources, it also leads to a natural greedy adaptive strategy, which can be driven in order to optimize the accuracy of PGD approximation [5,6]. We extend it to the cases of geometrical parameters models and PGD separation of space variables [7].
The focus of the talk is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds for those specific cases; (ii) error splitting and adaptive process when performing PGD-based model reduction.