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.
Mots-clés : computational fluid dynamics, optimization, uncertainty quantification, statistical inverse problems, data assimilation, hemodynamics
@article{M2AN_2013__47_4_1037_0,
author = {D{\textquoteright}Elia, Marta and Veneziani, Alessandro},
title = {Uncertainty quantification for data assimilation in a steady incompressible {Navier-Stokes} problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1037--1057},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
doi = {10.1051/m2an/2012056},
mrnumber = {3082288},
zbl = {1271.76062},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012056/}
}
TY - JOUR AU - D’Elia, Marta AU - Veneziani, Alessandro TI - Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1037 EP - 1057 VL - 47 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012056/ DO - 10.1051/m2an/2012056 LA - en ID - M2AN_2013__47_4_1037_0 ER -
%0 Journal Article %A D’Elia, Marta %A Veneziani, Alessandro %T Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1037-1057 %V 47 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012056/ %R 10.1051/m2an/2012056 %G en %F M2AN_2013__47_4_1037_0
D’Elia, Marta; Veneziani, Alessandro. Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1037-1057. doi : 10.1051/m2an/2012056. http://www.numdam.org/articles/10.1051/m2an/2012056/
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