Topics in Data Assimilation and Stochastic Partial Differential Equations

Abstract

In the first part of the thesis we consider data assimilation. We consider the nudging data assimilation algorithm applied to the 3D Navier-Stokes equations and we derive conditions, based entirely on the observations, that guarantee the global well-posedness, the regularity, and the tracking property of the nudging solution. Next we consider the nudging data assimilation algorithm applied to the 2D Navier-Stokes equations. We allow for random uncertainty in the model and we allow the observations to be contaminated with unbounded Gaussian error. We then do a computational study that compares the nudging and ensemble Kalman filter. Finally we consider the nudging algorithm applied to a planetary geostrophic viscous model for oceanic and atmosphere dynamics. In the second part of the thesis we studystochastic partial differential equations. We study the 2D stochastic Navier-Stokes equations with random viscosity. This is done with stochastic finite element discretizations and generalized polynomial chaos. Finally we study the linear stability of solutions to the 2D stochastic Navier-Stokes equations with random viscosity