What is the Lagrangian for Nonlinear Filtering?

Prashant Mehta, University of Illinois, USA - 11:15

There is a certain magic involved in recasting the equations in Physics, and the algorithms in Engineering, in variational terms.  The most classical of these ‘magics’ is the Lagrange’s formulation of the Newtonian mechanics.  An accessible modern take on all this and more appears in the February 19, 2019 issue of The New Yorker magazine: https://www.newyorker.com/science/elements/a-different-kind-of-theory-of-everything?reload=true

My talk is concerned with a variational (optimal control type) formulation of the problem of nonlinear filtering/estimation.  Such formulations are referred to as duality between optimal estimation and optimal control.  The first duality principle appears in the seminal (1961) paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic optimal control problem. 

In my talk, I will describe a generalization of the Kalman-Bucy duality theory to nonlinear filtering.  The generalization is an exact extension, in the sense that the dual optimal control problem has the same minimum variance structure for linear and nonlinear filtering problems. Kalman-Bucy’s classical result is shown to be a special case.  During the talk, I will also attempt to review other types of duality relationships that have appeared over the years for the problem of linear and nonlinear filtering.       

This is joint work with Jin Won Kim and Sean Meyn.  The talk is based on the following papers: https://arxiv.org/pdf/1903.11195.pdf and https://arxiv.org/pdf/1904.01710.pdf.

Invited by Jana de Wiljes