  We study time harmonic acoustic scattering on large deviation rough random
scatterers. Therein, the roughness of the scatterers is caused by a low Sobolev
regularity in the covariance function of their deformation field. The
motivation for this study arises from physical phenomena where small-scale
material defects can potentially introduce non-smooth deviations from a
reference domain. The primary challenge in this scenario is that the scattered
wave is also random, which makes computational predictions unreliable.
Therefore, it is essential to quantify these uncertainties to ensure robust and
well-informed design processes. While existing methods for uncertainty
quantification typically rely on domain mapping or perturbation approaches, it
turns out that large and rough random deviations are not satisfactory covered.
To close this gap, and although counter intuitive at first, we show that the
$p$-multilevel Monte Carlo method can provide an efficient tool for uncertainty
quantification in this setting. To this end, we discuss the stable
implementation of higher-order polynomial approximation of the deformation
field by means of barycentric interpolation and provide a cost-to-accuracy
analysis. Our considerations are complemented by numerical experiments in three
dimensions on a complex scattering geometry.
