Empowering PGD-based parametric analysis with Optimal Transport

The Proper Generalized Decomposition (PGD) is a Model Order Reduction framework that has been proposed to be able to do parametric analysis of physical problems. These parameters may include material properties, boundary conditions, etc. With this framework most of the computation may done in an off-line stage allowing to perform real time simulation in a variety of situations. Nevertheless, this scheme may lose its efficiency where the domain itself is also considered as "a parameter". Optimal transport techniques, on the other hand, have demonstrated exceptional performance in interpolating different types of fields described over geometrical domains with varying shapes. Hence trying allying both techniques is quite natural. The core idea is that PGD handles the parametric solution, while the optimal transport-based methodology transports the solution for a family of domains defined by geometrical parameters such as lengths, radii, thicknesses, etc. In this first attempt the associated methodology is proposed and apply in simple 1D and 2D cases showing interesting performances.