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.

中文翻译：

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双曲线守恒定律系统不确定性量化的侵入式加速策略

量化双曲线问题中不确定性影响的方法可以分为侵入式和非侵入式技术。非侵入式方法允许以黑盒方式使用给定的确定性求解器，同时令人尴尬地并行。但是，避免对给定求解器进行侵入性修改会剥夺使用多个固有侵入性数值加速工具的能力。此外，与非侵入式技术相比，侵入式方法有望以更少的未知数达到给定的精度。这种影响在具有高尺寸不确定性的环境中被放大。然而，侵入式方法的一个缺点是需要保证所产生的力矩系统的双曲线性。与随机伽辽金 (SG) 相比，侵入多项式矩 (IPM) 方法能够以解决每个空间单元和每个时间步中的优化问题为代价来保持双曲线性。在这项工作中，我们为侵入式方法提出了几种加速技术，并研究了它们与非侵入式随机搭配方法相比的优缺点。使用 IPM 求解稳态问题时，可以通过使用 PDE 约束优化的概念来降低重复求解 IPM 优化问题所产生的数值成本。此外，我们提出了 IPM 方法的自适应实现和高效并行化策略。对于流体动力学应用中的多维不确定性，证明了所提出的适应的有效性，导致在使用侵入式方法时需要较少数量的未知数来实现给定精度的观察结果。此外，使用所提出的加速技术，我们的实现比随机搭配更快地达到给定的精度。