Marine Ecosystem Model Calibration through Enhanced Surrogate-Based Optimization - Simulation and Modeling Methodologies, Technologies and Application - page 193

Fig. 3. Responses as in Figure 2, but for yet another section within the whole time interval. Again,
after employing the improvements in (13), the positive and negative peaks are removed (bottom).

negative response is non-physical and is a result of numerical errors due to using large
time steps in the numerical solution of the coarse model) and an upper bound a ub for
the correction factors. We furthermore restrict the correction factors to one in case the
fine and coarse model responses are below a certain threshold which should be of the
order of the discretization error below which the responses can be treated as zero.
More specifically, the following modifications of the model outputs and the scaling
factors are performed for each iteration k

0; a ub ;

if y c ≤ 0 if a k ≥ a ub

( i ) y c = , ( ii ) a k = ,

y c ; else a k ; else

(13)

β
f ≤ and y c ≤ ) ,

( iii ) a k =1 if ( y

where the operations are again meant point-wise and where (i) is applied before smooth-
ing. From numerical experiments, a ub =10 turned out to be a reasonable choice and
we furthermore consider =10 − 4 .
Figure 2 (bottom) shows the surrogate's, fine (down-sampled) and coarse model re-
sponse for the same illustrative tracer, time interval and depth layer, however, while
employing the improvements given in (13). It can be observed that the positive and
negative peaks present in the surrogate responses shown in Figure 2 (top) are removed
after applying (13). As additional evidence, Figure 3 (bottom) shows the same model
responses but for a different section within the whole time interval.
The numerical results presented in Section 5 demonstrate that this enhanced response
correction scheme allows us to further improve the computational efficiency of the SBO.