A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations
A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations
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双曲偏微分方程不确定性量化的新型弱侵入非线性多分辨率框架
DOI：
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发表时间：
2015
影响因子：
2.5
通讯作者：
G. Iaccarino
中科院分区：
文献类型：
--
作者：
G. Geraci;P. Congedo;R. Abgrall;G. Iaccarino
在本文中，一种新的多分辨率框架，即截断和编码（TE）方法，其先前由一些作者提出（Abgrall等人，J Comput Phys 257：19 - 56，2014）。doi：10.1016/j.jcp.2013.08.006），被推广和扩展以考虑偏微分方程（PDE）中的不确定性。创新的成分是由一个算法允许恢复的多分辨率表示，而不需要完全解决的解决方案，治疗的可能性，任何形式的PDF和使用高阶（甚至非线性，即数据依赖）重建的随机空间。此外，空间TE方法，这是一个弱侵入计划的不确定性量化（UQ），耦合的物理和随机空间，通过最小化计算成本的偏微分方程。所提出的计划是特别有吸引力的处理移动的不连续性（如激波在可压缩流），即使他们出现在模拟过程中，因为它是常见的非定常空气动力学应用。所提出的方法是非常灵活的，因为它可以很容易地耦合到不同的确定性计划，甚至与高分辨率的功能。本方法的灵活性和性能表现在各种数值测试情况下（代数函数和常微分方程），包括偏微分方程，线性和非线性，在随机性的存在。通过与求解UQ的一些经典方法（即Monte Carlo方法或非侵入多项式混沌方法）的比较，证明了该方法求解随机线性平流和Burgers方程的有效性.
In this paper, a novel multiresolution framework, namely the Truncate and Encode (TE) approach, previously proposed by some of the authors (Abgrall et al. in J Comput Phys 257:19–56, 2014. doi:10.1016/j.jcp.2013.08.006), is generalized and extended for taking into account uncertainty in partial differential equations (PDEs). Innovative ingredients are given by an algorithm permitting to recover the multiresolution representation without requiring the fully resolved solution, the possibility to treat a whatever form of pdf and the use of high-order (even non-linear, i.e. data-dependent) reconstruction in the stochastic space. Moreover, the spatial-TE method is introduced, which is a weakly intrusive scheme for uncertainty quantification (UQ), that couples the physical and stochastic spaces by minimizing the computational cost for PDEs. The proposed scheme is particularly attractive when treating moving discontinuities (such as shock waves in compressible flows), even if they appear during the simulations as it is common in unsteady aerodynamics applications. The proposed method is very flexible since it can easily coupled with different deterministic schemes, even with high-resolution features. Flexibility and performances of the present method are demonstrated on various numerical test cases (algebraic functions and ordinary differential equations), including partial differential equations, both linear and non-linear, in presence of randomness. The efficiency of the proposed strategy for solving stochastic linear advection and Burgers equation is shown by comparison with some classical techniques for UQ, namely Monte Carlo or the non-intrusive polynomial chaos methods.
影响因子：
4.1
作者：
Graham, I. G.;Kuo, F. Y.;Sloan, I. H.
通讯作者：
Sloan, I. H.