Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 399

w z

Measure

Noise

Fig. 6.3 In/Out system: z is the quantity measured, w is the quantity to be estimated

Noise i Noise m

Input

u ( k +1)

System Dynamics

Measure

z ( k +1)

Observability

k = k +1

Fig. 6.4 Possible approaches for the estimate with a system dynamics. Here we rely upon the
observability of the system and the knowledge of the statistical features of the stochastic process

modeling. In most of the cases, this is a system of partial differential equations and
a numerical discretization procedure is necessary for its quantitative solution. The
numerical model (i.e., the discretized mathematical model) reads

u .k C 1/ D u .k C 1/ .Past; Input; Noise/:

Assume that we measure z .k/ and we want to estimate u .k/ ( filtering problem)
or u .k C p/ (p-step prediction problem). If the variable u that we want to estimate
is a parameter of the original model (as it was the viscosity on the previous
example), this is an identification problem. In the simplest case, assume that the
system dynamics is linear and that noise affects both the input of the system and
the measures. The system dynamics is regarded as the mechanism converting the
p.d.f. of the input to the p.d.f. of the output. From the latter, we want to infer the
state variable. If we apply the criterion of finding the estimate by minimizing the
variance of the estimate, we will get a sequential estimate/prediction procedure
called Kalman filter . Although the method strongly relies on the linearity of the
system, that is not changing the nature of the p.d.f. , we will see that the method
can be properly extended to non-linear cases. The effectiveness of the procedure
relies upon the “observability” of the system, i.e., on how the output information is
actually representative of the state of the system (Fig. 6.4 ).
As pointed out, our goal here is to give a short introduction to probabilistic
estimation theory. For a more complete and extensive presentation, we refer to [ 68 ],
Chaps. 4-6.