较高维状态值的非线性滤波问题的实时算法研究

In this proposal, the principal investigator (PI) shall study the real-time algorithms of nonlinear filtering with medium high dimensional state. There are three topics covered in this proposal: the improved Carleman approach, the error estimate of solving Kushner or DMZ equation using the sparse grid algorithm, the on- and off-line solver of solving the path-wise robust DMZ equation with sparse grid algorithm and parallel computing. More precisely, PI proposes to augment the original state by its probabilists' Hermite polynomials as the new state in the improved Carleman approach. PI will derive the stochastic differential equation that the new state satisfies, and use the suboptimal method of the bilinear system to obtain the estimation of the new state, so get the original one. In the investigation of solving Kushner or DMZ equation with the sparse grid algorithm, PI suggests to consider the known results of the error estimation of the orthogonal projection onto the functional space spanned in certain sparse fashion, and analyzes the errors of solving the equations carefully. The on- and off-line algorithm to be adapted to solve the problems with medium high dimensional state is proposed to take the base functions with sparse grid on bounded domain, to numerically solve Kolmogorov forward equation accurately, and to use the parallel computing to perform a large amount of numerical integrations simultaneously, so that the unnormalized conditional density function of the state in the nonlinear filtering problems can be achieved in the real-time manner.

本项目中申请人将围绕较高维状态值的非线性滤波问题的实时算法展开研究工作，包括三个主要的探索方向：改进的卡莱曼逼近方法，稀疏网格对求解库什纳或DMZ方程的精确度影响，以及将稀疏网格算法和并行运算结合应用到线上线下运算结合的算法中实时求解较高维状态值的非线性滤波问题。更具体地说，在改进卡莱曼逼近方法中，申请人提出尝试通过原状态值的概率埃尔米特多项式扩展成新的状态值，进而推导出新状态值满足的随机微分方程。对该方程应用双线性系统的次最优算法可得到估计，从而得到原状态的估计。在利用稀疏网格求解库什纳或DMZ方程时，利用函数在稀疏网格形成的函数空间上的正交投影误差，细致分析其对谱方法求解方程时的误差影响。在线上线下运算结合的算法研究中，探讨有界区域上的基函数的稀疏网格，精确求解柯尔莫哥洛夫前向方程，并利用并行运算同时进行大量的数值积分，实时地逼近非线性滤波问题状态值的非规一化条件密度函数。

本项目中负责人及其合作者主要围绕较高维状态值的非线性滤波问题的算法展开研究工作。滤波问题在很多工程应用，包括目标跟踪、信号处理等问题中都有较广泛的应用。完成的研究内容包括：1、改进卡莱曼逼近方法；2、利用Askey类多项式混沌结合稀疏网格在逼近函数时的收敛性分析；3、高维双线性滤波问题的逼近算法及收敛性分析等。该项目取得了如下主要研究成果：1、发展了一个得到双线性滤波问题的线性最优估计，并在某些条件下，证明了该线性最优估计是在均方误差意义下的。2、将稀疏网格算法的逼近误差的证明推广到一般的Askey类正交多项式上，不仅仅局限于埃尔米特多项式。3、发展了一个结合埃尔米特多项式的改进卡莱曼逼近方法，可以给出非线性滤波问题的次最优估计。并数值验证了在精度上优于常用的扩展卡尔曼滤波。4、将线上线下算法进行进一步探索，在线下计算部分采用有界区域上的扩展雅可布正交多项式，证明了扩展雅可布谱方法求解向前Kolmogorov方程的收敛性。