Mathematics Colloquium
Date and time: Friday, 13 January, 2023, 4 pm
Venue: Ramanujan Hall
Host: Mayukh Mukherjee
Speaker: Prashant G. Mehta
Affiliation: University of Illinois at Urbana-Champaign, USA
Title: A variational formulation of nonlinear filtering
Abstract. There is a certain magic involved in recasting the equations in Physics in variational
terms. The most classical of these ‘magics’ is the Lagrange’s formulation of the Newtonian mechanics. An accessible take on all this and more appears in the February 19, 2019 issue of The New Yorker magazine. My talk is concerned with a variational (optimal control-type) formulation of the problem of nonlinear filtering/estimation. Such formulations are broadly referred to as duality between optimal estimation and optimal control. The first duality principle appears in the original (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 newly discovered generalization of the Kalman-Bucy duality to nonlinear filtering. As an application, I will discuss some comparisons between the stochastic stability of a Markov process and the (filter) stability of a conditioned process. Either of these is shown to arise from assuming a respective Poincaré inequality (PI). This is joint work with Jin Won Kim. The talk is based on the following papers:
https://arxiv.org/abs/2208.06586 and https://arxiv.org/abs/2208.06587