Lyapunov vectors and assimilation in the unstable subspace: theory and applications
Lyapunov vectors and assimilation in the unstable subspace: theory and applications
复制标题
不稳定子空间中的李雅普诺夫向量和同化：理论与应用
DOI：
10.1088/1751-8113/46/25/254020
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
A. Trevisan
中科院分区：
文献类型：
--
作者：
L. Palatella;A. Carrassi;A. Trevisan
基于有限数量的噪声观测，估计算法提供了当前时间系统状态的完整描述。在不稳定子空间（AUS）同化的名义下的估计算法利用其公式中的预测模型的非线性稳定性。通过将分析解限制在系统的不稳定和中性子空间中，有效地消除了由于对初始条件的敏感性而增长的误差，该子空间由具有正指数和零指数的李雅普诺夫向量所构成，而观测噪声不会沿着稳定方向干扰系统。本文综述了四维变分同化（4DVar-AUS）和扩展卡尔曼滤波（EKF-AUS）中的AUS方法及其在混沌模式中的应用。在这两种情况下，AUS算法至少是一样有效的，但更简单的实现和计算要求较低，比他们原来的同行。如理论所预测的，当误差动态是线性的时，4DVar-AUS的最佳子空间维度由正和零李雅普诺夫指数的数量给出，而EKF-AUS算法，使用相同的不稳定和中性子空间，恢复完整EKF算法的解，但是处理的误差协方差矩阵的维度要小得多，并且显著减少了计算负担。给出了大气环流简化模型和交通动力学最优速度模型的应用实例。本文是《物理学杂志A：数学与理论》特刊的一部分，专门讨论“李雅普诺夫分析：从动力系统理论到应用”。
Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models is reviewed. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. Examples of the application to a simplified model of the atmospheric circulation and to the optimal velocity model for traffic dynamics are given. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.