On the verification of PGD reduced-order models

International audience ; In current computational mechanics practice, multidimensional as well as multiscale or parametric models encountered in a wide variety of scientific and engineering fields often require either the resolution of significantly large complexity problems or the direct calculation of very numerous solutions of such complex models. In this framework, the use of model order reduction allows to dramatically reduce the computational requirements engendered by the increasing model complexity. Over the last few years, model order reduction techniques have sparked a growing interest in the whole scientific community [1, 2]. These appealing methods are based on separated variables representations of the solution of multi-parameter models lying in tensor product spaces. They enable to circumvent the terrific curse of dimensionality as the associated solution complexity scales linearly with the dimension of the tensor product space, whereas classical brute force approaches, such as mesh-based (or grid-based) approximation methods, face an exponentially growing solution complexity. Proper Generalized Decomposition (PGD) is currently one of the most popular Reduced-Order Modeling (ROM) techniques which can be interpreted as an extension of Proper Orthogonal Decomposition (POD). It allows the a priori construction of separated variables representations of the model solution without requiring any knowledge or information about this one (contrary to POD). Low-dimensional PGD reduced basis functions (or modes) are first constructed online on the fly by sequentially solving a series of few tractable simple problems. The resulting PGD-based approximate solution can then be computed offline, as it is defined explicitly in terms of all model parameters. Despite the good performances of PGD techniques, a major issue concerns the control of PGD reduced-order models and the development of robust and efficient verification tools able to assess the quality of PGD-based numerical approximations. A few works have been ...