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广义密度演化方程的δ函数序列解法

Solution of generalized density evolution equation via a family of δsequences

  • 摘要: 随机动力系统响应或状态向量的概率密度函数一般遵循概率密度演化方程,如Liouville方程、FPK方程和Dostupov-Pugachev方程,但是上述方程均属于高维偏微分方程,求解相当困难. 基于概率守恒原理的随机事件描述导出的广义密度演化方程,其维数与系统自由度无关,为随机动力系统分析提供了可能的途径. 从广义密度演化方程的形式解出发,引入δ函数的渐近序列,获得了广义密度演化方程的一种新的数值解法------广义密度演化方程的δ序列解法. 将建议方法与非参数密度估计进行了对比,指出非参数密度估计是该方法的一个特例. 最后,分别采用重构实例和演化实例验证了该方法在一维和多维情形下的有效性.

     

    Abstract: In the stochastic dynamics, it is one of the mostimportant purposes to acquire the probability density function and itsevolution process of the stochastic responses. The probability densityfunction of the responses or state vector of a stochastic dynamical systemis usually governed by some type of probability density evolution equationssuch as the Liouville, FPK or the Dostupov-Pugachev equation. However, theseequations in high dimension are too hard to obtain the solutions. Thegeneralized density evolution equation (GDEE), of which the dimension isindependent to the original dynamical system, provides a new possibility oftackling nonlinear stochastic systems. In this paper, based on the formalsolution of the GDEE, introducing the asymptotic sequences of the Diracδ function, a new numerical solution for the GDEE δ isproposed, to name as the Solution of GDEE via a family of δSequences. In addition, it is found that the non-parameter densityestimation can be regarded as a specific case of the proposed method. Atlast, the rationality and effectiveness of the proposed method is verifiedby some cases.

     

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