Solving First Order Linear Highly Underdetermined PDEs for Nonlinear Filters
Mr. Frederick Daum
1:30 - 2:30 PM, Tuesday, 13 November 2012
Shelby Center 160
We have invented a new nonlinear filter theory that is many orders of magnitude faster than standard particle filters for the same accuracy. Our theory uses particle flow (like physics) to compute Bayes’ rule, rather than a pointwise multiply. We construct a flow of the particles that is induced by a flow of the probability density using a log-homotopy. We do not use resampling of particles or proposal densities or importance sampling or any Markov chain Monte Carlo method. But rather, we design the particle flow with the solution of a linear first order highly underdetermined PDE, like the Gauss
divergence law in electromagnetics. We show a dozen distinct methods for solving this PDE, including: separation of variables, direct integration, method of characteristics and the Fourier transform. Our theory is related to Monge-Kantorovich optimal transport, but it is different in a number of ways. The talk explains what a particle filter is, and why engineers like particle filters, but we also explain the curse of dimensionality. We explain particle degeneracy and how we solve it with a simple cartoon. This talk is for normal mathematicians who do not have nonlinear filters for breakfast.