Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noisesGauss 与 Poisson 白噪声驱动下非线性系统解的概率密度
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Jia WT, Jiao Z, Zan WR, Zhu WQ, 2024. Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises. Probabilistic Engineering Mechanics, 77: 103658.DOI: 10.1016/j.probengmech.2024.103658
本研究提出了一种新方法，用于计算 Gauss 和 Poisson 白噪声组合激励下多维非线性动力系统解的概率密度。首先，根据随机系统的 Euler-Maruyama 格式和相应的 Chapman-Kolmogorov 方程推导出概率密度求解格式。该求解格式实际上是随机系统解的概率密度的显式数值表达。进而，我们为该求解格式提出了一种高效算法，实质上是数值积分实现。此外，我们证明了该求解格式是相应前向 Kolmogorov 方程的近似解。通过数值实例验证了我们提出的概率密度求解格式。关键词: 非线性随机系统, 概率密度, Gauss 和 Poisson 白噪声, Chapman-Kolmogorov 方程, 前向 Kolmogorov 方程, Fourier 变换A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.
Keywords: Nonlinear stochastic system; Probability density;Gaussian and Poisson white noises; Chapman–Kolmogorov equation; Forward Kolmogorov equation; Fourier transformation.![]()
图 1: Rayleigh-Van der Pol 振子求解格式构建的概率密度Fig. 1. Probability density constructed by the solver for (17)
图 2: 时刻为 5, 10, 20 时 Rayleigh-Van der Pol 振子求解格式与蒙特卡罗方法计算的边缘概率密度对比Fig. 2. Comparison of the marginal probability density computed by the solver and Monte Carlo method for (17) at t = 5, 10, 20
Fig. 3. L2-error
Fig. 4. Comparison of the marginal probability density computed by different methods
Fig. 5. L2-error
图 6: 时刻为 1, 3, 5, 7 时阻尼耦合振子求解格式计算的边缘联合概率密度Fig. 6. Marginal joint probability density for (18) computed by the solver at times t = 1, 3, 5, 7
图 7: 时刻为 5 时阻尼耦合振子的边缘概率密度云图: (a1-d1) 求解格式; (a2-d2) 蒙特卡罗方法Fig. 7. Contours of the marginal probability density for (18) at times t = 5: (a1-d1) Solver; (a2-d2) Monte Carlo Method
图 8: 时刻为 1, 3, 7 时阻尼耦合振子求解格式与蒙特卡罗方法计算的边缘概率密度对比Fig. 8. Comparison of the marginal probability density computed by the solver and the Monte Carlo method for (18) at t = 1, 3, 7
图 9: 求解格式与蒙特卡洛方法计算的边缘概率密度 Lebesgue 平方误差的演变Fig. 9. Evolution of the L2 errors for the marginal probability densities computed by the solver and the Monte Carlo method
图 10: 时刻为 0.5, 1, 1.5, 2 时高维系统的边缘联合概率密度Fig. 10. Marginal joint probability density p(t,x_1,x_3) for (19) at t = 0.5, 1, 1.5, 2
图 11: 时刻为 1 时高维系统的边缘概率密度云图: (a1-d1) 求解格式; (a2-d2) 蒙特卡罗方法Fig. 11. Contours of the marginal probability density for (19) at times t = 1: (a1-d1) Solver; (a2-d2) Monte Carlo Method
图 12: 时刻为 0.5, 1, 1.5 时高维系统求解格式与蒙特卡罗方法计算的边缘概率密度对比Fig. 12. Comparison of the marginal probability density computed by the solver and Monte Carlo method for (19) at t = 0.5, 1, 1.5
图 13: 求解格式与蒙特卡洛方法计算的边缘概率密度 Lebesgue 平方误差的演变Fig. 13. Evolution of the L2 errors for the marginal probability densities computed by the solver and the Monte Carlo method
作者信息 | Authors
西北工业大学 (Northwestern Polytechnical University) 数学与统计学院
焦哲 Zhe Jiao, 通讯作者 (Corresp.)西北工业大学 (Northwestern Polytechnical University) 数学与统计学院Email: zjiao@nwpu.edu.cn
西北大学 (Northwest University) 数学学院
浙江大学 (Zhejiang University) 力学系
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
