  In the last few years, several methods have been developed to deal with jump
singularities in parametric or stochastic hyperbolic PDEs. They typically use
some alignment of the jump-sets in physical space before performing well
established reduced order modelling techniques such as reduced basis methods,
POD or simply interpolation. In the current literature, the transforms are
typically of low resolution in space, mostly low order polynomials, Fourier
modes or constant shifts. In this paper, we discuss higher resolution
transforms in one of the recent methods, the transformed snapshot interpolation
(TSI). We introduce a new discretization of the transforms with an appropriate
behaviour near singularities and consider their numerical computation via an
optimization procedure.
