Beijing Institute of Mathematical Sciences and Applications

Jing Yan Li , Jie Wu , Nan Jun Yang

Xiao Pei Jiao

Monday, October 24, 2022 3:30 PM - 5:00 PM

1129B

Zoom 537 192 5549 (BIMSA)

滤波问题是指基于带有噪声的观测数据对系统的状态进行估计的问题。根据系统是否为线性，滤波问题可分为线性滤波和非线性滤波。滤波理论的核心问题是求解系统状态关于观测历史的条件概率密度函数$\rho(x,t)$。在最小均方误差的意义下，条件均值即为状态的最优估计。对于一般的非线性滤波系统，其非归一化的条件密度函数$\sigma(x,t)$满足著名的Duncan-Mortensen-Zakai (DMZ) 方程。这是一个随机偏微分方程，并且在一般情况下难以求得精确解。在20世纪70年代，Brockett，Clark和Mitter分别独立的通过引入估计代数的方法构造了有限维滤波器，并且得到了DMZ方程的精确解。我们将研究一类状态维数$n=3$，秩$r=2$的非最大秩估计代数，通过欠定偏微分方程等理论工具，建立了系统是有限维的充分必要条件。再利用结合代数工具，在任意维状态空间中构造了非最大秩的新型有限维滤波器。在该滤波器中，$\Omega$矩阵的元素可以是除了常数或多项式之外的一类$C^\infty$函数。

Xiaopei Jiao received his bachelor's degree from the Zhiyuan College of Shanghai Jiao Tong University and his Ph.D. from the Department of Mathematical Sciences at Tsinghua University. He subsequently worked as a postdoctoral researcher at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) and at the University of Twente in the Netherlands. His current research interests include finite-dimensional filtering theory, Yau-Yau filtering methods, physics-informed neural networks, and bioinformatics. His research focuses primarily on: (1) using geometric tools such as Lie algebras for solving partial differential equations and classifying nonlinear systems; (2) designing novel numerical algorithms based on physics-informed neural networks.