Abstract for Vincent Martinez

Vincent Martinez (Hunter College, CUNY)

Thursday, March 30, 2023

Parameter estimation in nonlinear PDEs

Abstract: In this talk, we will describe a class of algorithms for identifying unknown parameters of nonlinear PDEs. In the absence of observational errors, the convergence of these algorithms can be rigorously established under the assumption that sufficiently many scales of the solution are observed and that certain non-degeneracy conditions hold, which ensures identifiability of the parameters. This approach to parameter estimation is robust and can be applied not only to recover damping coefficients, but also external driving forces that are unknown apriori. Moreover, it is applicable to a large class of nonlinear equations, including many of those that arise in hydrodynamics, such as the 2D Navier-Stokes equations of incompressible flow, the 2D system for Rayleigh-Benard convection, the 3D primitive equations, or even dispersive-type models such as the 1D Korteweg-de Vries equation or 1D cubic nonlinear Schrödinger equation. We describe the derivation of these algorithms, address their convergence, and showcase the results of several computational experiments.