Biomedical Engineering Reference
In-Depth Information
reasonable to use the following cost functional
X
N
X
j
D
1
k
j;k
k
M
1
2
2
R
1
k
J
2
D
;
(6.72)
k
D
1
where
j;k
D
meas
.
x
j
;
k
/ .
x
j
;
k
/,R
k
is a weight s.p.d. matrix. Should
probabilistic information on the displacement be available, R
1
k
is the covariance
matrix of the noise of the displacement retrieval process.
As anticipated, we consider two approaches to solve this problem: a deterministic
variational approach and a Kalman-based approach.
Remark 6.8.
Typically, the time step t of the numerical scheme is smaller than
the time sample , requiring more observations than those available. A common
practice is to recover the observation at needed time steps by interpolation. In the
following we will use this approach.
Variational Approach
In order to minimize
J
1
we can use a gradient-based optimization approach as
discussed in Sect.
6.3.2
. However, as outlined there, the solution of an unsteady
minimization problem, such as the FSI problem, would be very expensive because
all the steps are coupled together, and it would also require the evaluation of shape
derivatives since the geometry is evolving in time. To reduce the computational
costs and the algorithm complexity, we exploit the fact that the parameter E does
not change in time and solve the following suboptimal problem. First, we discretize
the system in time. Then, at each time instant t
n
we solve a steady suboptimal
optimization problem, finding the value E
n
which minimizes the functional
Z
1
2
.
meas
.x;
n
/ .
x
;
n
//
2
d;
3
D
J
(6.73)
†
constrained by the time-discrete FSI problem at time t
n
. Finally, we compute E as
the average of E
n
: E D
N
P
n
D
1
E
n
.
1
Numerical Solution
For the sake of clarity, we focus on the simplified membrane model (
6.70
), already
discretized in time. However, note that the optimization strategy described in the
following has been applied to the original FSI problem (
6.69
)in[
59
]. When
considering the membrane approximation, the cost functional J
3
becomes
Z
meas
n
2
d;
1
2
n
J
m
D
†
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