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. On the other hand, intrusive modifications allow for certain acceleration techniques. 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 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. Integrating the iteration from the numerical treatment of the optimization problem into the moment update reduces numerical costs, while preserving local convergence. 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.

量化不确定性在双曲线问题中的影响的方法可以分为介入技术和非介入技术。非侵入式方法允许以黑盒的方式使用给定的确定性求解器，同时使它们尴尬地并行。另一方面，侵入式修改允许某些加速技术。而且，与非侵入性技术相比，预期侵入性方法将以较少的未知数达到给定的精度。在具有高尺寸不确定性的设置中，这种影响会放大。侵入式方法的缺点是需要保证所得力矩系统的双曲性。与随机加勒金（SG）相比，

在这项工作中，我们提出了几种用于介入方法的加速技术，并研究了与非介入式随机配置方法相比的优缺点。当使用IPM解决稳定问题时，可以通过使用PDE约束优化中的概念来减少重复解决IPM优化问题而产生的数字成本。将优化问题的数值处理中的迭代集成到力矩更新中，可以减少数值成本，同时又保持局部收敛性。此外，我们提出了IPM方法的自适应实现和有效的并行化策略。对于流体动力学应用中的多维不确定性，证明了所提出的改编方法的有效性，结果发现，使用侵入性方法时，需要较少数量的未知数才能达到给定的精度。此外，使用建议的加速技术，我们的实现比随机搭配更快地达到了给定的精度。