Explicit Convergence Analyses of PDE-Based Filtering Algorithms

.The direct method and the Yau–Yau algorithm are two crucial filtering methods in nonlinear filtering problems. Recently, a series of research works [X. Luo and S.-T. Yau, IEEE Trans. Automat. Control, 58 (2013), pp. 2563–2578; X. Luo and S.-T. Yau, IEEE Trans. Automat. Control, 58 (2013), pp. 2495–2507; Z. Wang, X. Luo, S. S.-T. Yau, and Z. Zhang, IEEE Trans. Automat. Control, 65 (2019), pp. 1613–1624; W. Dong, X. Luo, and S. S.-T. Yau, IEEE Trans. Automat. Control, 66 (2021), pp. 1559–1572; S. Li, Z. Wang, S. S.-T. Yau, and Z. Zhang, IEEE Trans. Automat. Control, 68 (2023), pp. 4405–4412; X. Chen, J. Shi, and S. S.-T. Yau, IEEE Trans. Automat. Control, 64 (2019), pp. 1648–1654; J. Shi, X. Chen, W. Dong, and S. S. T. Yau, IEEE Control Systems Lett., 1 (2017), pp. 233–237] in these two methods have gained great success in nonlinear numerical experiments. Therefore, a key issue is whether a general framework of convergence analyses can be developed for these numerical algorithms based on partial differential equations. The contributions of this work consist of two parts. The first one is that we set up a convergence analysis framework for the direct method and the Yau–Yau algorithm, which can show the deep connections between these two methods. They are different formulations of solving the robust DMZ equation by reducing it to solve the Yau–Yau PDE [B. Balaji, PMC Phys. A, 3 (2009), pp. 1–28]. The second one is that we prove the explicit convergence rates of the Yau–Yau algorithm with the spectral method and the direct method with Gaussian approximation, in terms of the numbers of spectral basis and time steps.