[hal-01057246] Robust control of PGD-based reduced models

Nowadays, numerical simulation constitutes a common tool in science and engineering activities; it is especially used for prediction, decision making, or simply for a better understanding of physical phenomena. However, in order to give an accurate representation of the real world, a large set of parameters may need to be introduced in the mathematical models involved in the simulation, which leads to a huge number of degrees of freedom (due to the so-called curse of dimensionality) if classical brute force methods are employed. In this context, model reduction methods are necessary in order to avoid important (and often overwhelming) computational efforts. During the last few years, appealing model reduction techniques, based on separation of variables within a spectral resolution approach (such as the POD), have received a growing interest. In particular, a technique called Proper Generalized Decomposition (PGD) has been very recently introduced as a POD extension. Contrary to the POD, the PGD approximation does not require any knowledge on the solution (it is thus referred as a priori ); it operates in an iterative strategy in which a set of simple problems, that can be seen as pseudo eigenvalue problems, need to be solved. However, even though the PGD is usually very effective, a major drawback is that it does not include, until now, any robust error estimate that could give an idea of the quality of the approximate solution. In the present work, we introduce a consistent error estimator for numerical simulations performed by means of the Proper Generalized Decomposition (PGD) approximation. This estimator, which is based on the constitutive relation error, enables to capture all error sources (i.e. those coming from space and time numerical discretizations, from the truncation of the PGD decomposition, etc. . . ) and leads to guaranteed bounds on the exact error. The specificity of the associated method is a double approach, i.e. a kinematic approach and a unusual static approach, for solving the parameterized problem by means of PGD. This last approach makes straightforward the computation of a statically admissible solution, which is necessary for robust error estimation. An attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes; it thus represents a practical and relevant tool for driving algorithms carried out in PGD, being possibly used as a stopping criterion or as an adaptation indicator. Numerical experiments on multi-parameter elasticity or transient thermal problems illustrate the performances of the proposed method for global and goal-oriented error estimation.



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