  Methods for quantifying the effects of uncertainties in hyperbolic problems
can be divided into intrusive and non-intrusive techniques. Non-intrusive
methods allow the usage of a given deterministic solver in a black-box manner,
while being embarrassingly parallel. However, avoiding intrusive modifications
of a given solver takes away the ability to use several inherently intrusive
numerical acceleration tools. Moreover, intrusive methods are expected to reach
a given accuracy with a smaller number of unknowns compared to non-intrusive
techniques. This effect is amplified in settings with high dimensional
uncertainty. A downside of intrusive methods is however the need to guarantee
hyperbolicity of the resulting moment system. In contrast to
stochastic-Galerkin (SG), the Intrusive Polynomial Moment (IPM) method is able
to maintain hyperbolicity at the cost of solving an optimization problem in
every spatial cell and every time step. In this work, we propose several
acceleration techniques for intrusive methods and study their advantages and
shortcomings compared to the non-intrusive Stochastic Collocation method. When
solving steady problems with IPM, the numerical costs arising from repeatedly
solving the IPM optimization problem can be reduced by using concepts from
PDE-constrained optimization. Additionally, we propose an adaptive
implementation and efficient parallelization strategy of the IPM method. The
effectiveness of the proposed adaptations is demonstrated for multi-dimensional
uncertainties in fluid dynamics applications, resulting in the observation of
requiring a smaller number of unknowns to achieve a given accuracy when using
intrusive methods. Furthermore, using the proposed acceleration techniques, our
implementation reaches a given accuracy faster than Stochastic Collocation.
