Separated Representations & PGD Based Model Reduction: Fundamentals and Applications
July 8, 2013 — July 12, 2013
- Francisco Chinesta (Ecole Centrale de Nantes, France)
- Pierre Ladeveze (LMT/ENS Cachan/CNRS/UPMC/PRES, UniverSud Paris, France)
Today many problems in science and engineering remain intractable, in spite of the impressive progresses attained in modelling, numerical analysis, discretization techniques and computer science during the last decade. In fact their numerical complexity, or the restrictions imposed by different requirements (real-time on deployed platforms, for instance) make them unaffordable for today’s technologies. We can enumerate different challenging scenarios for efficient numerical simulations:
(i) models defined in high-dimensional spaces suffering the so-called curse of dimensionality, usually encountered in quantum chemistry, kinetic theory descriptions, chemical master equation, …;
(ii) simulation based real time control;
(iii) multi-scale and multi-physics non-linear problems involving strong couplings;
(iv) models defined in degenerated domains; (v) problems needing too many direct solutions (optimization, inverse analysis, …);
(vi) DDDAS (dynamic data driven application systems);
(vii) augmented reality needing fast simulation in deployed computing platforms and finally;
(viii) models involving uncertainty.
While the previous list is by no means exhaustive, it includes a set of problems with no apparent
relationship between them that can, however, be treated in a unified manner as will show this course. Their common ingredient is our lack of capabilities (or knowledge) to solve them numerically in a direct, traditional way. In order to obtain a solution, some kind of model order reduction is thus compulsory.
In this course we first describe the construction of reduced models by revisiting POD (Proper Orthogonal Decomposition) and reduced bases models, from the point of view of their mathematical foundations and some challenging applications. Then, we will move to a new generation of simulation strategies based on the use of separated representations (space – including physical and conformational coordinates -, space-time or space-time-parameters) at the origin of the so-called Proper Generalized Decomposition – PGD – techniques.
Because such representation allows circumventing the course of dimensionality, models can be enriched by considering parameters as extra-coordinates, making possible fast and cheap online calculations from off-line richer parametric solutions.
Separated representations will be analyzed in detail, from their mathematical foundations to their most spectacular applications. We will illustrate how this approximation could constitute a new paradigm in computational sciences allowing to circumvent the above computational issues in a panoply of applications in engineering sciences, as the ones just referred.
The course is addressed to doctoral students, young and senior researchers, practicing engineers working in the area of simulation software, who are faced by the strong limitations of standard simulation techniques for solving complex models, and require new approaches for ensuring efficient simulations of such challenging models.