Abstract:
Motivated by the many real world systems exhibiting non-smooth dynamics, we will explore in this talk the definition of a sequential estimator for a state variable governed by a variational inequality, up to perturbation, for which past observations are known, up to perturbation. A first approach is to relax the boundary constraint of the synthetic variable (the solution of the v.i.) to build an approximated variant of the Mortensen estimator that uses the resulting nonlinear smooth dynamics.
But a second path leads to a surprising result! Contrary to the case of smooth dynamics, here the zero-noise limit of the robust form of the Zakai equation (that governs an unnormalised conditional density associated to the state variable given the observations) cannot be understood from the Bellman equation of the value function arising in Mortensen’s deterministic estimation: a deterministic estimation problem. It can, however, be associated with a control problem. This results in a loss of equivalence between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.
Biography:
Álvaro Mateos González is a Postdoctoral Instructor of Mathematics at NYU Shanghai. He holds a PhD in mathematics from the École Normale Supérieure de Lyon, where he obtained his BA and Master's degrees, and the Inria Rhône-Alpes. He conducts research in mathematics inspired by biological systems and other applications. His main topics of interest are the asymptotic analysis of integro-partial differential equations, Hamilton-Jacobi equations, numerical analysis, stochastic variational inequalities, and mathematical modelling in biology: anomalous diffusion, brain-bacteria interactions, and non-smooth systems.
He is currently a member of the Modeling, Data Analysis and Computational Methods in Biology Research Group based at Univ. Complutense of Madrid and Univ. of Torino, and a participant in two grants from the National Science Foundation of China: Modeling, Diffusion Approximation, and Stochastic Control of Nonsmooth Systems - PI: Laurent Mertz; and Convection and fluid-structure interactions in geological phase transitions - PI: Mac Huang.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
This event is open to the NYU Shanghai community and Math community.