Marine Ecosystem Model Calibration through Enhanced Surrogate-Based Optimization - Simulation and Modeling Methodologies, Technologies and Application - page 190

value or a specific number of iterations (particularly if the computational budget of the
optimization process is limited).
If the surrogate

s k satisfies so-called zero-order and first-order consistency condi-
tions with the fine model at u k , i.e.,

s k ( u k )= y f ( u k ) , s k ( u k )= y f ( u k ) ,

(8)

s k ( u k ) denote the derivatives of the responses, the surrogate-based scheme
(7) is provable convergent to at least a local optimum of (5) under mild conditions
regarding the coarse and fine model smoothness, and provided that the surrogate opti-
mization scheme (7) is enhanced by the trust-region (TR) safeguard, i.e.,

y and

with

u k +1 =argmin
u ∈U ad ,
u − u k ≤δ k

J ( s k ( u )) ,

(9)

with δ k being the trust-region radius updated according to the TR rules. We refer the
reader to e.g. [4,8] for more details.

4.1 Surrogate Model Using Basic Multiplicative Response Correction

It has been found in [14] that a natural way of constructing the surrogate would be multi-
plicative response correction . This approach is motivated by the fact that the qualitative
relation of the fine and coarse model response is rather well preserved (at least locally)
while moving from one parameter vector to another. As a result, a multiplicative correc-
tion allows constructing a surrogate model with a good generalization capability. The
technique is briefly recalled below.
The surrogate response

s k ( u ) , at iteration k of the optimization process, is generated
by multiplicative correction of the smoothed coarse model response, denoted by y c ,
which we briefly formulate as

⎫
⎬

s k ( u ):= a k y c ( u ) ,

k =1 , 2 ,...
β = M f /M c

(10) β

f ( u k )
y c ( u k )

y

⎭

a k :=

where the operations in (10) are meant point-wise and where a k denote the correction
factors which are included in the vector a k . They are defined as the point-wise divi-

sion of the smoothed and down-sampled fine model response, denoted by y

β
f ,bythe

smoothed coarse model response at the point u k .

It was observed that smoothing allows us to remove the numerical noise from the
coarse model response and identify the main characteristics of the traces of interest (see
[14] for details). The fine model response is smoothed accordingly in the formulation
(10).
Down-sampling was necessary to make the fine model response commensurable
with the corresponding response of the coarse model. The down-sampled fine model
response

β
f is simply given as
y ji := y βj,i , j =1 ,...,M c , i =1 ,...,I.

y

(11)