This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition-PGD. Space and time separated representations generalize Proper Orthogonal Decompositions-POD-avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions, . . .) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

International audienceThis paper revisits a powerful discretization technique, the Proper Generalized Decomposition-PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, aEuro broken vertical bar but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, aEuro broken vertical bar) opening numerous possibilities (optimization, inverse identification, real time simulations, aEuro broken vertical bar)

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to highdimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with highdimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.

Standard simulation in classical mechanics is based on the use of two very different types of equations. The first one, of axiomatic character, is related to balance laws (momentum, mass, energy,...), whereas the second one consists of models that scientists have extracted from collected, natural or synthetic data. Even if one can be confident on the first type of equations, the second one contains modeling errors. Moreover, this second type of equations remains too particular and often fails in describing new experimental results. The vast majority of existing models lack of generality, and therefore must be constantly adapted or enriched to describe new experimental findings. In this work we propose a new method, able to directly link data to computers in order to perform numerical simulations. These simulations will employ axiomatic, universal laws while minimizing the need of explicit, often phenomenological, models. This technique is based on the use of manifold learning methodologies, that allow to extract the relevant information from large experimental datasets.

Engineering is evolving in the same way than society is doing. Nowadays, data is acquiring a prominence never imagined. In the past, in the domain of materials, processes and structures, testing machines allowed extract data that served in turn to calibrate state-of-the-art models. Some calibration procedures were even integrated within these testing machines. Thus, once the model had been calibrated, computer simulation takes place. However, data can offer much more than a simple state-of-the-art model calibration, and not only from its simple statistical analysis, but from the modeling and simulation viewpoints. This gives rise to the the family of so-called twins: the virtual, the digital and the hybrid twins. Moreover, as discussed in the present paper, not only data serve to enrich physically-based models. These could allow us to perform a tremendous leap forward, by replacing big-data-based habits by the incipient smart-data paradigm.