Events for 11/02/2021 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Vincent Martinez, City University of New York Hunter College

Title: Convergence analysis for a parameter estimation algorithm for the 2D Navier-Stokes equations

Abstract: In this talk, we will discuss the problem of the determination of unknown parameters in a dynamical system via observations on a subset of the systems’s state variables. In our setup, we consider a 2D incompressible viscous fluid whose kinematic viscosity is assumed to be unknown, but its filtered velocity field is observed continuously-in-time. The algorithm studied is one proposed by Carlson, Hudson, and Larios (2020) in which a feedback control paradigm for data assimilation of PDEs, originally introduced by Azouani, Olson, and Titi (2014), is leveraged to provide systematic updates to the value of the viscosity based on the collected observations. Although several computational tests had been carried out that corroborate the convergence of this algorithm to the true value of viscosity, a rigorous proof remained elusive. We discuss a direct proof of convergence in the regime of observing a sufficiently large resolution of the velocity field, under the assumption that a certain non-degeneracy condition holds. The two main ingredients of the proof are the availability of higher-order sensitivity-type bounds and identification of a Lyapunov-type structure for the time-derivative of the energy of the velocity error.