Generalized Jacobi Spectral Method in Solving Nonlinear Filtering Problems

The on-and off-line algorithm for the nonlinear filtering (NLF) problems was developed by Yau and the first author, and the Hermite spectral method (HSM) has been implemented to serve as the off-line computations. Notice that the true states in real applications can always be assumed to be bounded. In this paper, we shall investigate the Jacobi spectral method (JSM) instead to numerically solve the forward Kolmogorov equation (FKE) arising in NLF problems. The convergence rate of JSM to FKE is analyzed in the suitable function space, which is twice as fast as that in HSM. The formulation has been detailed for an essentially infinite-dimensional NLF problem, the 1-d cubic sensor problem. Compared with HSM, the JSM yields more accurate result.