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5
Numerical Results
For all optimization runs, we use the MATLAB
1
function
fmincon
, exploiting the
active-set algorithm. The following cost functions
I
M
c
(
z
ji
−
(
y
d
)
ji
)
2
,
J
(
z
):=
z
−
y
d
2
=
(14)
i
=1
j
=1
M
c
I
J
(
z
):=
z
−
y
d
2
=
(
z
ji
−
(
y
d
)
ji
)
2
,
(15)
i
=1
j
=1
were the target data - as a test case - is given by model generated, attainable data as
β
f
(
u
d
)
.
y
d
:=
y
For the optimization runs presented in this paper we employ the following cost func-
tions: for the fine model optimization, we use (14) with
β
f
, for the coarse model
z
=
y
z
=
y
c
and for the SBO, (15) with
optimization, (15) with
z
=
s
k
, whereas (14)
was used in the termination condition and to compare the results and where the down-
sampled fine model response
β
f
is defined by (11). Sampling was necessary to yield
a comparable fine model optimization run while in (15) the smoothed target data is
considered accordingly, since the coarse model and thus also the surrogate's response
are smoothed. Note that the cost functions we employ are not normalized by the total
number of discrete model points. The dimension of the responses is of the order of
10
5
. Clearly, this has to be taken into account for presented cost function values in the
following.
We perform an exemplary direct fine and coarse model optimization as well as a SBO
based on the surrogate in (10) exploiting the original and improved response correction
scheme (cf. Sections 4.1, 4.3). In the following, the solutions of the four optimization
runs are compared through visual inspection of the (down-sampled) fine model response
y
y
β
f
)
(cf. (14)) at the respective optima.
The optimization cost is measured in
equivalent
fine model evaluations which are
determined taking into account the coarsening factor
β
. More specifically, one evalu-
ation of the coarse model with a coarsening factor
β
is equivalent to
1
/β
evaluations
of the fine model. On the other hand, the cost of one iteration of the SBO (in terms of
equivalent fine model evaluations) equals to the number of coarse model evaluations
necessary to optimize the surrogate model divided by this factor
β
, and increased by
the cost for the response correction. Recall that the specific correction (10) we use in
this paper requires one fine model evaluation only.
Figure 4 shows the value of the cost function
J
(
y
β
f
and the corresponding cost function value
J
(
y
β
f
)
versus the equivalent number of
fine model evaluations for the SBO algorithm using the surrogate model exploiting the
original and the improved correction scheme, as well as for the fine and coarse model
optimization. Points 1 and 3 in Figure 4 indicate those solutions obtained in the SBO
1
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