Determining White Noise Forcing From Eulerian Observations in the Navier Stokes Equation

TL;DR

This paper develops a Bayesian framework for inferring white noise forcing in the 2D Navier-Stokes equations from noisy velocity observations, establishing theoretical continuity and demonstrating numerical posterior analysis.

Contribution

It introduces a Bayesian inverse problem approach with a Gaussian process prior for the forcing, proving posterior well-posedness and continuity with respect to data.

Findings

Posterior distribution is absolutely continuous w.r.t. the prior.

Posterior is a continuous function of the observed data.

Numerical simulations validate the theoretical results.

Abstract

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data. We complement this theoretical result with numerical simulation of the posterior distribution.