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.

非线性系统的最佳连续离散滤波需要演化前向 Kolmogorov 方程，即 Fokker-Planck 方程，与贝叶斯条件更新交替。我们提出了两种数值网格方法来表示网格或网格上的密度函数。对于低维、平滑的系统，有限体积法是一种有效的求解器，可以给出收敛到最佳连续时间值的估计。我们给出了数值例子来表明这种有限体积滤波器能够处理由秩亏观测引起的多模态滤波分布，并且可以在滤波过程中执行贝叶斯最优参数估计。有限体积滤波器中使用的密度函数的朴素离散导致计算成本和存储随着维度的增加呈指数增长，这使得有限体积滤波器对于高维问题不可行。我们通过使用密度函数的张量序列表示（或近似值）来规避这种“维数诅咒”，并采用对张量序列表示进行操作的高效隐式 PDE 求解器。我们给出了跟踪的数值例子n 个连续时间的弱耦合钟摆，以演示具有多达 80 维的复杂密度函数的过滤。