  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. As reduced basis methods, it
relies on snapshots for the reconstruction of parameter dependent functions so
that it is efficiently applicable in a PDE context. However, we allow a
transformation of the physical coordinates before the use of a snapshot in the
reconstruction, which allows to realign 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
1d and 2d numerical experiments.
