Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 455

Fig. 6.9 On the left the synthetic measures generated adding Gaussian noise to an analytical
solution are reported on a layer crossing the axis of symmetry. On the right , on a layer close to
the one where the measures are collected, the assimilated velocity is reported. Adapted from [ 40 ],
with permission of c Elsevier 2010

that the noise affects the boundary data (whereas in [ 18 ] they are considered exact),
which is always the case in real applications. For the numerical solution with the
WLSFE method the boundary and internal data are properly weighted according
to the noise level (assumed known in this particular experiment); the assimilated
solution, on an internal layer close to the measurement one, is reported in Fig. 6.9
(right). The filtering action of the DA on the noise is evident. Quantitative analysis,
not reported here, reveals a good level of accuracy [ 40 ].

WLSFEM as a Bayesian Approach to DA

In [ 21 ] a reinterpretation of the WLSFEM in terms of Bayesian approach to DA
is proposed; in fact, in [ 40 ] the method is not presented in an inverse problem
framework. Here we show that the WLSFE solution can be interpreted as the
maximum a posteriori (MAP) estimator in a variational Bayesian approach to DA,
for a certain choice of a priori distribution and likelihood function. A statistical
interpretation of the weights is also provided.
In describing the method we refer to the general boundary value problem ( 6.65 ).
We recall that in a Bayesian approach to DA all variables are treated as random,
the goal is to determine the p.d.f. of u conditioned on realizations of the measures
d 1 ;:::;d N s available on the internal layers 1 ;:::; N s . We assume that the
measures are affected by the measurement noise 1 ;:::; N s such that d i . x / D
u . x /j i C. x / i ,fori D 1;:::;N s . To apply the Bayes theorem we need to define an
a priori distribution for u , p u , based on our prior belief on u and a likelihood function
for the measurement noise i , p ;i . In order to show the equivalence between the
WLSFE deterministic solution, or WLSFE estimator, and the MAP estimator in the
Bayesian setting we make the following choices.
We define a prior distribution which is large when u satisfies the governing
equations ( 6.65 ) “well” and small otherwise; in this way the prior describes to what
extent the equations are a good model for the observations. Formally

p u . u / / exp f J. u / g ;

where J is defined as in ( 6.66 ).
Next, in defining the likelihood functions for i , we assume that the measurement
errors i are independent and normally distributed with null mean and variance