Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation

Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.