Intrusive acceleration strategies for Uncertainty Quantification for hyperbolic systems of conservation laws
Intrusive acceleration strategies for Uncertainty Quantification for hyperbolic systems of conservation laws
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DOI：
10.1016/j.jcp.2020.109698
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发表时间：
2019-12
期刊：
影响因子：
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通讯作者：
J. Kusch;J. Wolters;M. Frank
中科院分区：
文献类型：
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作者：
J. Kusch;J. Wolters;M. Frank
双曲问题中不确定性影响的量化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，同时是并行的。另一方面，侵入式修改允许某些加速技术。此外，与非侵入性技术相比，侵入性方法预计将以较少的未知数达到给定的精度。这种效应在具有高维度不确定性的设置中被放大。侵入式方法的缺点是需要保证所得到的矩系统的双曲性。与随机Galerkin（SG）方法相比，插入式多项式矩（IPM）方法能够以在每个空间单元和每个时间步长上求解优化问题为代价来保持双曲性，本文提出了几种插入式方法的加速技术，并研究了它们与非插入式随机配置方法相比的优缺点。当用IPM求解稳态问题时，通过使用PDE约束优化的概念，可以减少重复求解IPM优化问题所产生的数值代价。将来自优化问题的数值处理的迭代集成到矩更新中降低了数值成本，同时保持局部收敛。此外，我们还提出了IPM方法的自适应实现和高效并行化策略。所提出的适应性的有效性证明了流体动力学应用中的多维不确定性，导致在使用侵入式方法时，需要较少数量的未知数，以达到给定的精度的观察。此外，使用建议的加速技术，我们的实现达到给定的精度比随机搭配更快。
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.