PGD-Based Computational Vademecum for Efficient Design, Optimization and Control

Abstract

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.

A Space-Time PGD-DIC Algorithm:

“…As an alternative, the purpose of this work is to build a resolution scheme based on the use of some variable separation techniques widely encountered in numerical simulations [2,25,19,12,10,42]. The interest of extending such methods to DIC has already been illustrated in previous works [27,17], where the dimensions of space were separated in order to speedup computation time.…”

Real-time simulation techniques for augmented learning in science and engineering

“…Instead, PGD provides a practical means of obtaining meta-models for complex scenarios. These meta-models or computational vademecums [26] are then at real-time feedback rates.…”

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…Following our former works (see [20,5,3,21] and the references therein) one could consider model parameters (as well as boundary conditions, initial conditions or geometrical parameters) as extra-coordinates, in order to calculate a parametric solution. Thus, for example, in order to compute the solution of problem (1) for any value of the diffusion coefficient k ∈ Ω k , it suffices considering k as another coordinate (like x or y) and looking for u(x, y, k) as described in [17].…”