Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities
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间歇性不稳定系统中不确定性量化的多项式混沌的基本局限性
DOI：
10.4310/cms.2013.v11.n1.a3
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发表时间：
2013
影响因子：
1
通讯作者：
A. Majda
中科院分区：
文献类型：
--
作者：
M. Branicki;A. Majda
本文研究了截断多项式混沌展开式（PCE）和截断GramCharlier展开式（GrChE）作为间歇性和正Lyapunov指数非线性系统的不确定性量化（UQ）方法的适用性。这两种方法依赖于截断的伽辽金投影，或者是系统变量在一个固定的多项式基上跨越“不确定”子空间（PCE），或者是与系统的不确定演化（GrChE）相关的联合概率分布的合适的特征函数展开。基于一个简单的，统计上精确可解的非线性和非高斯检验模型，我们详细说明了利用截断谱展开的方法，无论是PCE还是GrChE，对于具有间歇性不稳定性或阻尼参数不确定性的系统的不确定性量化有明显的局限性。间歇性和厚尾概率密度是湍流惯性和耗散范围的标志特征，我们表明，在如此重要的动力体系中，PCE的表现至多类似于作者在经验信息论框架内的UQ的不同背景下使用的简单得多的高斯力矩闭合技术。此外，我们还表明，GrChE近似的不可实现性与动力学中的间歇性发作有关，并且经常伴随着二阶统计量在短时间内的错误爆炸。这两种截断谱展开的局限性来自于以下几点：(i) PCE和GrChE在时间上的非均匀收敛，导致随着时间的推移，良好逼近随机过程所需的项数迅速增加；（ii）由于相关Wiener过程的光谱表示的有限截断，导致高斯白噪声强迫导致捕获随机性的恒定通量的基本问题；（iii）间歇性存在时PCE和GrChE系数的缓慢衰减；阻碍了稀疏截断方法的实现，而稀疏截断方法已广泛应用于近椭圆问题或低雷诺数流动。通过利用简单的非线性和非高斯但统计上精确可解的测试模型的直接测试，可以充分说明这些限制的严格证明，该测试模型在这里被提出作为间歇性系统中UQ算法的具有挑战性的基准。
Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated GramCharlier Expansions (GrChE) as possible methods for uncertainty quantification (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents. These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the ‘uncertain’ subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE). Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties in the damping. Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory. Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times. These limitations of the two types of truncated spectral expansions arise from the following: (i) Non-uniform convergence in time of PCE and GrChE resulting in a rapidly increasing number of terms necessary for a good approximation of the random process as time evolves, (ii) Fundamental problems with capturing the constant flux of randomness due to white Gaussian noise forcing via finite truncations of the spectral representation of the associated Wiener process, (iii) Slow decay of PCE and GrChE coefficients in the presence of intermittency, hampering implementation of sparse truncation methods which have been widely used in nearly elliptic problems or in low Reynolds number flows. Rigorous justification of these limitations is richly illustrated by straightforward tests exploiting a simple nonlinear and non-Gaussian but statistically exactly solvable test model which is proposed here as a challenging benchmark for algorithms for UQ in systems with intermittency.