A Homotopy Method for Grid Based Nonlinear Filtering

With increasing calculation power of modern systems, the focus of Stochastic Filtering turns to nonlinear effects. Sophisticated methods have to be investigated in application areas, where linearized methods like the Extended Kalman Filter tend to sub-optimality or even divergence. Fully nonlinear solutions to the estimation problem are provided by an approximation of the full probability density function (pdf) in particle filters or the Fokker-Planck equation. Both methods suffer from the degeneration of the approximated probability density function caused by the application of pure Bayes rule for the measurement update. For particle filters the particle flow successfully overcomes this problem. Unfortunately the particle flow can not be directly adapted to grid based methods which solve the Fokker-Planck equation. In this contribution a new approach to solve the problem of degeneration for grid based nonlinear filtering methods is presented by introducing a grid flow concept. It consists of a common flow of the whole grid, which preserves the underlying grid structure and is supplemented by a compensation step, which considers the measurement effects, that cannot be treated by a common flow. The advantages of this grid flow approach are shown for a seven-dimensional nonlinear tracking example, which is solved by the Fokker-Planck equation on sparse grids. It turns out, that the grid flow approach increases the estimation accuracy in comparison to pure Bayes rule.