Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem

Dept. of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, USA.
mdelia2@mathcs.emory.edu; ale@mathcs.emory.edu

Received: 24 December 2011

Abstract

The reliable and effective assimilation of measurements and numerical simulations in engineering applications involving computational fluid dynamics is an emerging problem as soon as new devices provide more data. In this paper we are mainly driven by hemodynamics applications, a field where the progressive increment of measures and numerical tools makes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusion of noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose is the quantification of uncertainty affecting velocity and flow related variables of interest, all treated as random variables. The method consists in the solution of an optimization problem where the misfit between data and velocity - in a convenient norm - is minimized under the constraint of the NSE. We derive classical point estimators, namely the maximum a posteriori – MAP – and the maximum likelihood – ML – ones. In addition, we obtain confidence regions for velocity and wall shear stress, a flow related variable of medical relevance. Numerical simulations in 2-dimensional and axisymmetric 3-dimensional domains show the gain yielded by the introduction of a complete statistical knowledge in the assimilation process.