[hal-01057248] Goal-oriented control of PGD-based simulations

During the last few years, due to the increasing number of multi-parameter simulation models, model reduction techniques have been the object of a growing interest in research and industry. In particular, an appealing technique based on separation of variables within a spectral resolution approach was recently introduced and successfully used in many applications of Computational Mechanics [1]. It is called Proper Generalized Decomposition (PGD). Contrary to the classical POD, the PGD approximation does not require any knowledge on the solution; it operates in an iterative strategy in which basis functions (or modes) are computed on the fly, by solving simple pseudoeigenvalue problems. However, even though the PGD is very effective for a large class of problems, a major question is to derive verification tools in order to control the quality of the approximate PGD solution. A verification approach was introduced in [3] to build guaranteed PGD-reduced models for linear elliptic or parabolic problems depending on parameters. This approach, which is based on the concept of constitutive relation error and adjoint techniques, provides for strict bounds on the error on outputs of interest. It also enables to assess contributions of various error sources (space and time discretizations, truncation of the PGD decomposition, etc.), which can help driving adaptive strategies. In this work, we present the new advances which have been performed in this goal-oriented PGD-verification method. They particularly enable to deal with problems with lots of parameters. Furthermore, they aim at setting up a non-intrusive procedure in order to address implementation issues. Therefore, virtual charts associated with quantities of interest and computed from PGD models can now benefit fully from the verification method to satisfy a prescribed accuracy.



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