SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 731-762, January 2021.
We develop a novel computational method for evaluating the extreme excursion probabilities arising from random initialization of nonlinear dynamical systems. The method uses a Markov chain Monte Carlo or a Laplace approximation approach to construct a biasing distribution that in turn is used in an importance sampling procedure to estimate the extreme excursion probabilities. The prior and likelihood of the biasing distribution are obtained by using Rice's formula from excursion probability theory. We use Gaussian mixture biasing distributions and approximate the non-Gaussian initial state distribution by the method of moments to circumvent the linearity and Gaussianity assumptions needed by excursion probability theory. The method requires the tangent linear model of the nonlinear dynamical system to be bounded. Additionally, our method works best when the uncertainty-induced trajectories of the nonlinear dynamical system are small perturbations around a mean linear trajectory and thus when the target system exhibits only mild nonlinearity. We demonstrate the effectiveness of this computational framework for nonlinear dynamical systems of up to 100 dimensions.

中文翻译：

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通过莱斯公式有效计算动力系统极端偏移概率

SIAM/ASA 不确定性量化杂志，第 9 卷，第 2 期，第 731-762 页，2021 年 1 月。
我们开发了一种新的计算方法来评估由非线性动力系统的随机初始化引起的极端偏移概率。该方法使用马尔可夫链蒙特卡罗或拉普拉斯近似方法来构建偏置分布，该偏置分布又用于重要性采样程序以估计极端偏移概率。偏差分布的先验和似然是通过使用偏移概率理论中的莱斯公式获得的。我们使用高斯混合偏置分布，并通过矩方法近似非高斯初始状态分布，以规避偏移概率理论所需的线性和高斯假设。该方法要求非线性动力系统的切线线性模型有界。此外，当非线性动力系统的不确定性引起的轨迹是围绕平均线性轨迹的小扰动时，我们的方法效果最好，因此当目标系统仅表现出轻微的非线性时。我们证明了这种计算框架对于多达 100 维的非线性动力系统的有效性。