Approximating the linear response of physical chaos
Approximating the linear response of physical chaos
复制标题
近似物理混沌的线性响应
作者：
Adam A. Śliwiak;Qiqi Wang
统计量的参数导数是预测、设计优化和不确定性量化中非常需要的量。在存在混沌的情况下，这些量的严格计算当然是可能的，但在数学上很复杂，计算起来也很昂贵。基于Ruelle的形式，证明了复杂的线性响应算法可以在具有物理空间统计齐性的高维系统中得到极大简化。我们认为，如果目标函数与不稳定的流形适当地对齐，则SRB(Sinai-Ruelle-Bowen)度量梯度的贡献可以忽略不计，而SRB(Sinai-Ruelle-Bowen)度量梯度是完整算法中最复杂的积分部分。无论目标函数和扰动参数的物理意义和数学形式如何，现实世界中的一大类混沌系统都有可能满足这一抽象条件。我们展示了几个支持这些结论的数值例子，并展示了简化的线性响应算法的使用和性能。在数值实验中，我们考虑了由微分方程组描述的物理模型，包括Lorenz 96和Kuramoto-Sivashinsky。
Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.