Regularity estimate and sparse approximation of pathwise robust Duncan-Mortensen-Zakai equation

Abstract

In this paper, we establish an \textit{a priori} estimate for arbitrary-order derivatives of the solution to the pathwise robust Duncan-Mortensen-Zakai (DMZ) equation within the framework of weighted Sobolev spaces. The weight function, which vanishes on the physical boundary, is crucial for the \textit{a priori} estimate, but introduces a loss of regularity near the boundary. Therefore, we employ the Sobolev inequalities and their weighted analogues to sharpen the regularity bound, providing improvements in both classical Sobolev spaces and H{ö}lder continuity estimates. The refined regularity estimate reinforces the plausibility of the quantized tensor train (QTT) method in [S. Li, Z. Wang, S. S.-T. Yau, and Z. Zhang, IEEE Trans. Automat. Control, 68 (2023), pp. 4405--4412] and provides convergence guarantees of the method. To further enhance the capacity of the method to solve the nonlinear filtering problem in a real-time manner, we reduce the complexity of the method under the assumption of a functional polyadic state drift $f$ and observation $h$. Finally, we perform numerical simulations to reaffirm our theory. For high-dimensional cubic sensor problems, our method demonstrates superior efficiency and accuracy in comparison to the particle filter (PF) and the extended Kalman filter (EKF). Beyond this, for multi-mode problems, while the PF exhibits a lack of precision due to its stochastic nature and the EKF is constrained by its Gaussian assumption, the enhanced method provides an accurate reconstruction of the multi-mode conditional density function.