# Joint AMS-ECE Seminar

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**Gromov’s method for particle flow filters**

##### Fred Daum

Raytheon

Tuesday, 4/24/18, 2:00pm

Light Engineering 250

**Abstract:** We derive a new exact stochastic particle flow filter, using a theorem of Gromov.
Our filter is many orders of magnitude faster than standard particle filters for high
dimensional problems, and our filter beats the extended Kalman filter accuracy by
orders of magnitude for difficult nonlinear problems. Our theory uses particle flow
to compute Bayes’ rule, rather than a pointwise multiply. We do not use resampling
of particles or proposal densities or any MCMC method. But rather, we design the particle
flow with the solution of a linear first order highly underdetermined PDE. We solve
this PDE as an exact formula which is valid for arbitrary smooth nowhere vanishing
densities. Gromov proves that there exists a “nice” solution to a linear constant
coefficient PDE for smooth functions if and only if the number of unknowns is sufficiently
large (at least the number of linearly independent equations plus the dimension of
the state vector). A “nice” solution of the PDE means that we do not need any integration,
and hence it is very fast. To dispel the mystery of Gromov’s theorem we show the simplest
non-trivial example. We also show several generalizations of Gromov’s theorem. Particle
flow is similar to optimal transport, but it is much simpler and faster because we
avoid solving a variational problem. Optimal transport is (almost always) deterministic
whereas our particle flow is stochastic, like all such algorithms that actually work
robustly for difficult high dimensional problems.

**Bio:** Fred Daum is an IEEE Fellow, a Principal Fellow at Raytheon, a Distinguished Lecturer
for the IEEE and a graduate of Harvard University. Fred's exact fixed finite dimensional
nonlinear filter theory generalizes the Kalman and Beneš filters. Fred’s particle
flow nonlinear filter is many orders of magnitude faster than standard particle filters
for the same accuracy. He has published more than one hundred technical papers, and
he has given invited lectures at MIT, Harvard, Yale, Caltech, the Technion, Ecole
Normale Superieure de Paris, Brown, Georgia Tech., Duke, Melbourne Univ., Univ. of
Toulouse, Univ. of New South Wales, Univ. of Canterbury, Liverpool Univ., Xidian Univ.,
Univ. of Illinois at Chicago, Washington Univ. at St Louis, etc.