  The numerical approximation of solutions of parametric or stochastic
hyperbolic PDEs is still a serious challenge. Because of shock singularities,
most methods from the elliptic and parabolic regime, such as reduced basis
methods, POD or polynomial chaos expansions, show a poor performance. Recently,
Welper [Interpolation of functions with parameter dependent jumps by
transformed snapshots. SIAM Journal on Scientific Computing, 39(4):A1225-A1250,
2017] introduced a new approximation method, based on the alignment of the jump
sets of the snapshots. If the structure of the jump sets changes with
parameter, this assumption is too restrictive. However, these changes are
typically local in parameter space, so that in this paper, we explore $h$ and
$hp$-adaptive methods to resolve them. Since local refinements do not scale to
high dimensions, we introduce an alternative "tensorized" adaption method.
