Model Order Reduction in High-dimensional Parameter Space
Model reduction techniques such as Proper Orthogonal Decomposition, Proper Generalized Decomposition and Reduced Basis are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the « curse of dimensionality ». The today practical limit is about a dozen parameters and much less for nonlinear engineering problems.
The talk will introduce a general answer to this challenge . This MOR method named the « parameter-multiscale PGD » is based on the Saint-Venant’s Principle which works for numerous models in Physics. Such a principle highlights two different levels of parametric influence, which drives us to introduce a multiscale description of the parameters and to separate a « macro » and a « micro » scale, as it is classically done for space or time. A first implementation of this vision has been done using a Discontinuous Galerkin spacial formulation and applied to an elasticity 3D-problem composed of up to a thousand parameters. Last developments concern its extension to classical finite element solvers and nonlinear problems .
The talk will give the basic features of the « parameter-multiscale PGD », especially its mechanical bases, and presents its extensions. Elasticity 3D- problems will be used to illustrate its performance as well as its current limits. The latest developments and perspectives will also be shown.
 Ladeveze P., Paillet C., Neron D. (2018) Extended-PGD Model Reduction for Nonlinear Solid Mechanics Problems Involving Many Parameters. In: Oñate E., Peric D., de Souza Neto E., Chiumenti M. (eds) Advances in Computational Plasticity. Computational Methods in Applied Sciences, vol 46. Springer, Cham
Ladeveze P. On reduced models in nonlinear solid mechanics, European Journal of Mechanics A/Solids 60: 227-237, 2016.