关于信号与噪声相关的非线性滤波问题的分解算法研究

The principal investigator (PI) proposes to study the splitting-up methods of two simplified nonlinear filtering models with correlated noises based on Duncan-Mortensen-Zakai (DMZ) equation and feedback particle filter (FPF) respectively. To be more specific, we shall study (1) the splitting-up method of solving DMZ equation. The solution of DMZ equation will be approximated by two recursive processes: the one is a deterministic process, satisfying a second-order parabolic partial differential equation, while the other one is a stochastic process, satisfying a first-order degenerate stochastic differential equation. The key of the decomposition is that the deterministic process should be independent of the online observation data. Besides the convergence analysis of the splitting-up method, the PI proposes to solve the deterministic process beforehand using the idea in the on- and off-line algorithm and to give a computable real-time algorithm. (2) the splitting-up method of FPF. Firstly, the PI shall derive the Euler-Lagrange (E-L) equations corresponding to the two simplified models with correlated noises. The E-L equations of the gain functions in FPF are obtained by minimizing the Kullback-Leibler divergences. Secondly, we shall numerically solve the gain function off-line by the similar idea in on- and off-line algorithm, so that the efficiency of the on-line computation in FPF will be improved. Lastly, the PI shall analyze the error introduced in the splitting-up procedure and show the convergence of the splitting-up FPF.

申请人提出对两类简化的噪声相关模型分别从基于Duncan-Mortensen-Zakai（DMZ）方程和反馈粒子滤波探讨分解算法。具体地说，（1）基于DMZ方程的分解算法。从Trotter算子分解思想出发将DMZ方程分解成两个过程：一个是确定性过程，其满足二阶抛物偏微分方程；另一个则是随机过程，其满足一阶退化的随机微分方程。分解的关键在确定性过程中不包含在线观测。除理论证明该分解算法的收敛性，申请人还将分解算法结合线上线下计算思想把确定性过程的求解挪至线下，给出有效的实时求解算法。（2）反馈粒子滤波的分解算法。首先根据极小化Kullback-Leibler散度推导出简化噪声相关模型的反馈粒子滤波中的增益函数满足的欧拉-拉格朗日方程。再将增益函数的求解结合线上线下思想挪至线下提前计算，以提高反馈粒子滤波的在线计算效率。最后申请人对分解引入的误差进行估计，并证明该分解算法的收敛性。

非线性滤波是工程与应用数学交叉领域的重要问题，有广泛的应用前景。本项目中针对状态和观测噪声相关的非线性滤波问题从不同的数学思想方法提出几种有效的分解算法。完成的研究结果如下：①理论证明基于Duncan-Mortensen-Zakai（DMZ）方程的噪声相关非线性滤波问题的分解算法收敛性；②形式化给出了噪声相关非线性滤波问题的反馈粒子滤波算法；③联系基于Kushner方程的分解算法给出反馈粒子滤波算法的理论严格推导。这些理论算法的研究将为后续的算法设计提供理论依据和保障。