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 [1]. 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 [2].

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.

[1] 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

[2]Ladeveze P. On reduced models in nonlinear solid mechanics, European Journal of Mechanics A/Solids 60: 227-237, 2016.