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5
years
= 43800
discrete time step. We will furthermore use the subscript
f
to distin-
guish the relevant fine model variables, which read
y
f
,τ
f
and
M
f
, from those we will
introduce for the coarse model, respectively.
2.3 The Low-Fidelity Model
Marine ecosystem model, are typically given as coupled time-dependent partial differ-
ential equations, compare [5,17]. One straightforward way to introduce a low-fidelity
(or coarse) model for these models is to reduce the spatial and/or temporal resolution,
whereas, in this paper, we exploit the latter one.
The coarse model, which is a less accurate but computationally cheap representation
of
y
f
is obtained by using a coarser time discretization with a discrete time step
τ
c
given as
τ
c
=
βτ
f
,
(3)
with a
coarsening factor
β ∈
N
\{
0
,
1
}
, while keeping the spatial discretization fixed.
The state variable for this coarser discretized model will be denoted by
y
c
, the corre-
sponding number of discrete time steps by
M
c
=
M
f
/β
, i.e., we have
M
c
I
, I
=
n
z
n
t
.
(
y
c
)
j
=((
y
c
)
ji
)
i
=1
,...,I
, j
=1
,...,M
c
,
y
c
∈
R
(4)
Note that the parameters
for this model are the same as for the fine one.
Clearly, the choice of the temporal discretization, or equivalently, the coarsening
factor
β
, determines the quality of the coarse model and of a surrogate if based upon
the latter one. Moreover, both the computational cost, the performance and quality of
the solution obtained by a SBO process might be affected.
Altogether, we seek for a reasonable trade-off between the accuracy and speed of the
coarse model. From numerical experiments, a value of
β
=40
turned out be a reason-
able choice, as was shown in [14]. Numerical results presented in Section 4 demonstrate
that such a coarse model leads to a reliable approximation of the original fine ecosys-
tem model when a response correction technique as described in this paper is utilized.
Furthermore, it was observed that, for this specific choice of
β
, while additionally re-
stricting the parameter
u
1
, i.e., the sinking velocity, using an appropriate upper bound,
the resulting model response does not show any numerical instabilities.
u
3
Optimization Problem
The task of parameter optimization in climate science typically is to minimize a least-
squares type cost function measuring the misfit between the discrete model output
y
=
y
(
u
)
and given observational data
y
d
[2,21]. In most cases, the problem is constrained
by parameter bounds. The optimization problem can generally be written as
u
∈U
ad
J
(
y
(
u
))
,
min
(5)
where
J
(
y
):=
||
y
−
y
d
||
2
,
(6)
n
n
,
b
l
<
b
u
.
U
ad
:=
{
u
∈
R
:
b
l
≤
u
≤
b
u
},
b
l
,
b
u
∈
R
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