Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.
Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.
复制标题
线性和非线性统计响应理论以及复杂湍流动力系统的灵敏度分析和统计控制的原型应用。
作者：
A. Majda;D. Qi
统计响应理论为涉及大量未解决的不稳定模式的高维复杂湍流系统（例如在气候变化科学中）的分析和统计预测提供了有效的工具。最近，线性和非线性响应理论通过识别最敏感的响应方向，在克服湍流系统的不确定量化和统计控制中的维数灾难方面取得了有希望的发展。我们提供了在原型模型层次结构下使用统计响应理论解决各种具有挑战性的问题的广泛说明，从简单的可解方程到各向异性地球物理湍流。直接应用线性响应算子进行统计响应被证明对于小扰动范围仅具有有限的技能。对于更强的非线性和扰动，保证模型保真度和灵敏度的非线性降阶统计模型降阶策略提供了一个系统框架来恢复前导统计中的多尺度变异性。线性响应算子应用于训练阶段，以获得仅需要未扰动平衡统计的最佳非线性模型响应。统计响应理论进一步应用于固有高维系统的统计控制。平均值的统计响应提供了一种从统计能量方程恢复控制力的有效方法，而无需运行昂贵的模型。在所有测试示例中，统计响应策略在具有不同统计特征的各种动态机制中显示出统一的鲁棒技能。统计响应理论的进一步应用包括预测湍流被动传输中的极端事件和间歇性，以及控制初始和外部不确定性的总统计增长的严格饱和界限。
Statistical response theory provides an effective tool for the analysis and statistical prediction of high-dimensional complex turbulent systems involving a large number of unresolved unstable modes, for example, in climate change science. Recently, the linear and nonlinear response theories have shown promising developments in overcoming the curse-of-dimensionality in uncertain quantification and statistical control of turbulent systems by identifying the most sensitive response directions. We offer an extensive illustration of using the statistical response theory for a wide variety of challenging problems under a hierarchy of prototype models ranging from simple solvable equations to anisotropic geophysical turbulence. Directly applying the linear response operator for statistical responses is shown to only have limited skill for small perturbation ranges. For stronger nonlinearity and perturbations, a nonlinear reduced-order statistical model reduction strategy guaranteeing model fidelity and sensitivity provides a systematic framework to recover the multiscale variability in leading order statistics. The linear response operator is applied in the training phase for the optimal nonlinear model responses requiring only the unperturbed equilibrium statistics. The statistical response theory is further applied to the statistical control of inherently high-dimensional systems. The statistical response in the mean offers an efficient way to recover the control forcing from the statistical energy equation without the need to run the expensive model. Among all the testing examples, the statistical response strategy displays uniform robust skill in various dynamical regimes with distinct statistical features. Further applications of the statistical response theory include the prediction of extreme events and intermittency in turbulent passive transport and a rigorous saturation bound governing the total statistical growth from initial and external uncertainties.