Interpolation of Functions with Parameter Dependent Jumps by Transformed Snapshots

Functions with jumps and kinks typically arising from parameter dependent or stochastic hyperbolic PDEs are notoriously difficult to approximate. If the jump location in physical space is parameter dependent or random, standard approximation techniques like reduced basis methods, PODs, polynomial chaos, etc., are known to yield poor convergence rates. In order to improve these rates, we propose a new approximation scheme. It relies on snapshots for the reconstruction of parameter dependent functions so that it is efficiently applicable in a PDE context. However, the physical domain of each individual snapshot is transformed before its use in the reconstruction, which allows realignng the moving discontinuities and yields high convergence rates. The transforms are automatically computed by minimizing a training error. In order to show feasibility of this approach, it is tested by numerical experiments with one and two physical dimensions and one parameter dimension.