Dr. F.E. Daum
Senior Principal Fellow, Raytheon Company, USA
We have invented a new theory of exact particle flow for nonlinear filters. This generalizes our theory of particle flow that is already many orders of magnitude faster than standard particle filters and which is several orders of magnitude more accurate than the extended Kalman filter for difficult nonlinear problems. The new theory generalizes our recent log-homotopy particle flow filters in three ways: (1) the particle flow corresponds to the exact flow of the conditional probability density corresponding to Bayes’ rule; (2) roughly speaking, the old theory was based on incompressible particle flow (like subsonic flight in air), whereas the new theory allows compressible flow (like supersonic flight in air); (3) the old theory suffers from obstruction of particle flow as well as singularities in the equations for flow, whereas the new theory has no obstructions and no singularities. Moreover, our basic filter theory is a radical departure from all other particle filters in three ways: (a) we do not use any proposal density; (b) we never resample; and (c) we compute Bayes’ rule by particle flow rather than as a point wise multiplication. We have made hundreds of numerical experiments to test this new theory, using several classes of examples: quadratic & cubic nonlinearities of the measurements, stable & unstable dynamical systems, linear systems, multimodal probability densities, and radar tracking problems. It turns out that the computational complexity of particle filters (for optimal accuracy) depends on the following parameters, which we vary in our numerical experiments: dimension of the state vector of the plant, stability or instability of the plant (as gauged by the eigenvalues of the plant), initial uncertainty of the state vector of the plant, signal-to-noise ratio of the measurements, and the process noise of the plant. Particle filters generally suffer from the curse of dimensionality, whereas our filter substantially mitigates this effect for the examples studied so far, for plants with dimension up to 30. Other particle filters generally do not exploit any smoothness or structure of the problem, whereas our new theory assumes that the densities are twice continuously differentiable in x (the state vector of the plant), and we assume that the densities are nowhere vanishing. We design the particle flow using the solution of a first order linear (highly underdetermined) PDE, like the Gauss divergence law in electromagnetics. We analyze 11 methods for solving this PDE, including an exact solution for certain special cases (Gaussian and exponential family), solving Poisson’s equation, separation of variables, generalized method of characteristics, direct integration, Gauss’s variational method, optimal control, generalized inverse for linear differential operators, and another homotopy inspired by Gromov’s h-principle. The key issues are: (1) how to select a unique solution; (2) stability of the particle flow; and (3) computational complexity to solve the PDE. This talk is for normal engineers who do not have log-homotopy for breakfast.
ההרצאה תתקיים ביום רביעי 13.10.2010
שעה 16:30
בנין אוירונוטיקה חדר 241
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