IMPROVED SURROGATE-BASED OPTIMIZATION OF A MARINE

ECOSYSTEM MODEL USING RESPONSE CORRECTION

M. Prieß

, S. Koziel

and T. Slawig

Institute for Computer Science, Cluster The Future Ocean, Christian-Albrechts Universit¨at zu Kiel, 24098 Kiel, Germany

Engineering Optimization & Modeling Center, School of Science and Engineering, Reykjavik University

Menntavegur 1, 101 Reykjavik, Iceland

Keywords:

Climate models, Marine ecosystem models, Surrogate-based optimization, Parameter optimization, Response

correction.

Abstract:

An improved surrogate-based optimization (SBO) methodology is developed for the optimization of climate

model parameters. Our technique is based upon a multiplicative response correction technique to create a

surrogate from a temporarily coarser discretized physics-based low-ﬁdelity model. The original version of

this methodology was successfully applied to calibration of a (one-dimensional) representative of a class of

marine ecosystem models yielding about 84% computational cost savings when compared to the high-ﬁdelity

model optimization. Here, we demonstrate that by employing relatively simple modiﬁcations of the response

correction scheme, the surrogate model accuracy and the efﬁciency of the optimization process can be further

improved. More speciﬁcally, for the considered test case, the optimization cost is reduced three times when

compared to the original technique, i.e., from about 15% to only 5% of the cost of the direct high-ﬁdelity

ecosystem model optimization (used as a benchmark method). The corresponding time savings are increased

to 95%.

1 INTRODUCTION

Surrogate-based optimization (SBO) (Queipo et al.,

2005) is a methodology to efﬁciently optimize com-

plex, so-called high-ﬁdelity models, that require sub-

stantial computational effort already for a model eval-

uation. High-ﬁdelity models are typically evaluated

through computer simulation and evaluation times of

several hours, days or even weeks are not uncommon.

As a consequence, optimization and control problems

for such models are often still beyond the capability

of modern numerical algorithms and computer power.

The idea of SBO is to exploit a surrogate, a computa-

tionally cheap and yet reasonably accurate representa-

tion of the high-ﬁdelity model. The surrogate replaces

the high-ﬁdelity model in the optimization process in

the sense of providing predictions of the model opti-

mum. Also, it is updated using the high-ﬁdelitymodel

data accumulated during the process. The prediction-

updating scheme is normally iterated in order to reﬁne

the search and to locate the high-ﬁdelity model opti-

mum as precisely as possible. One of possible ways of

creating the surrogate, our work in this paper is based

on, is to utilize a physically-based low-ﬁdelity model.

SBO is widely and very successfully used in engi-

neering sciences (Bandler et al., 2004; Forrester and

Keane, 2009; Leifsson and Koziel, 2010; Queipo

et al., 2005). The application on parameter optimiza-

tion in climate models is new.

Climate models are typically given as time-

dependent partial differential or differential algebraic

equations (PDEs/DAEs) (Majda, 2003; McGufﬁe and

Henderson-Sellers, 2005; Gill, 1982). One example

are marine ecosystem models (Fennel and Neumann,

2004; Sarmiento and Gruber, 2006), one of which

our work in this paper is based on. Marine ecosys-

tem models describe photosynthesis and other bio-

geochemical processes in the marine ecosystem that

are important, e.g., to compute and predict the oceanic

uptake of carbon dioxide (CO

) as part of the global

carbon cycle (Sarmiento and Gruber, 2006). Since the

number of processes that have to be included and the

needed temporal and spatial resolution is quite high,

so is the computational effort.

The aim of parameter optimization is to adjust or

identify the model parameters such that the model re-

sponse ﬁts given measurement data (Banks and Ku-

nisch, 1989). The mathematical task thus can be clas-

siﬁed as a least-squares type optimization or inverse

449

Prieß M., Koziel S. and Slawig T..

IMPROVED SURROGATE-BASED OPTIMIZATION OF A MARINE ECOSYSTEM MODEL USING RESPONSE CORRECTION.

DOI: 10.5220/0003597704490457

In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SDDOM-2011), pages

449-457

ISBN: 978-989-8425-78-2

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

problem (Tarantola, 2005).

This optimization (or calibration) process requires

a substantial number of (typically expensive) function

and optionally sensitivity or gradient or even Hessian

matrix evaluations. Hence, decreasing the effort re-

lated to the function evaluations (or, equivalently, cut-

ting down the number of function calls necessary to

ﬁnd the optimum) is of primary importance to reduce

the overall optimization cost. This becomes particu-

larly signiﬁcant for computationally expensive three-

dimensional coupled models, for example, global cli-

mate models (Gill, 1982).

In (Prieß et al., 2011), a surrogate-based method-

ology has been developed for the optimization of

climate model parameters. The technique is based

upon a multiplicative response correction technique

to create a surrogate from a temporarily coarser dis-

cretized physics-based low-ﬁdelity model. It has been

successfully applied to a (one-dimensional) represen-

tative of a class of marine ecosystem models and

demonstrated to yield substantial savings of the com-

putational cost of the optimization process when com-

pared to a direct optimization of the high-ﬁdelity

model.

In this paper, we demonstrate that by employing

simple modiﬁcations of the original response correc-

tion scheme, one can improve the surrogate’s accu-

racy, as well as further reduce the computational cost

of the optimization process. We verify our approach

by using synthetic target data and by comparing the

results of SBO with the improved surrogate to those

obtained with the original one. The optimization cost

is reduced three times when compared to previous re-

sults, i.e., from about 15% to only 5% of the cost of

the direct high-ﬁdelity ecosystem model optimization

(used as a benchmark method). The corresponding

time savings are increased to from 84% to 95%.

The paper is organized as follows. The high-

ﬁdelity ecosystem model, considered here as a test

problem, as well as a low-ﬁdelity counterpart that we

use as a basis to construct the surrogate model, are de-

scribed in Section 2. The optimization problem under

consideration is formulated in Section 3. The orig-

inal and improved response correction schemes and

the comparison of the corresponding surrogate model

qualities are discussed in Section 4. Numerical results

for an illustrative SBO run are provided in Section 5.

Section 6 concludes the paper.

2 MODEL DESCRIPTION

The considered example for a climate model is a one-

dimensional marine ecosystem model (Oschlies and

Garcon, 1999) driven by pre-computed ocean circula-

tion data. In the following, we brieﬂy describe the

high-ﬁdelity model and its low-ﬁdelity counterpart

which is a basis to construct a surrogate for further

use in the optimization process.

2.1 The High-ﬁdelity Model

Simulating the marine ecosystem has become a key

tool for understanding the ocean carbon cycle and its

variability. The marine ecosystem contains several

biogeochemical quantities (called tracers), for exam-

ple nutrients, phyto- and zooplankton which inter-

act and are moreover transported by the ocean cir-

culation and inﬂuenced by temperature and salinity.

Thus, ecosystem simulations require modeling and

computation of both ocean circulation and biogeo-

chemistry. The underlying continuous models are

governed by coupled systems of nonlinear, parabolic

PDEs or DAEs, for ocean circulation (ocean models,

i.e., Navier-Stokes equations with additional temper-

ature and salinity transport equations) and transport

of biogeochemical tracers (marine ecosystem models,

i.e., convection- or advection-diffusion-reaction type

equations) (Sarmiento and Gruber, 2006).

In ecosystem models, the parameters to be opti-

mized – in the following summarized in the vector u

– are, for example, growth and dying rates of the trac-

ers and thus appear in the usually nonlinear coupling

or interaction terms in the model.

0 2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

1.2

1.4

time [ hours ]

Detritus [ mmol N m

−3

]

y(u)

Figure 1: Model response y

(D)

(detritus) and observation

data y

(D)

for one year at depth z ≃ −25m.

Our example ecosystem model was developed by

Oschlies and Garcon (1999)and simulates the interac-

tion of dissolved inorganic nitrogen, phytoplankton,

zooplankton and detritus (dead material) – thus also

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

450

called NPZD model. One aim was to reproduce ob-

servations y

at different North Atlantic locations by

the optimization of model parameters within credible

limits. Figure 1 shows the model response and target

data, respectively, as illustration for the tracer detritus

for a certain depth and a part of the time interval.

The model uses pre-computed ocean circulation

and temperature data from an ocean model (in a some-

times called off-line modus), i.e., no feedback by the

biogeochemistry on the circulation and temperature is

modeled (Oschlies and Garcon, 1999).

Since biogeochemistry happens locally and sink-

ing processes only in the vertical water column, we

use here a one-dimensional version of the model at

a given horizontal position. This is additionally mo-

tivated by the fact that there have been special time

series studies at ﬁxed locations. Clearly, the compu-

tational effort of a one-dimensional simulation is sig-

niﬁcantly smaller than for the three-dimensional case.

Thus, since the one-dimensional case includes all sig-

niﬁcant features of ecosystem models, it can serve as

a good veriﬁcation example for testing the applica-

bility of surrogate-based approaches that can be later

exploited for optimizing the 3D model.

In the NPZD model, the concentrations (in

mmol N m

−3

) of dissolved inorganic nitrogen N, phy-

toplankton P, zooplankton Z, and detritus D are sum-

marized in the vector y = (y

(l)

)

l=N,P,Z,D

and described

by the following coupled PDE system

∂y

(l)

∂t

∂

∂z

∂y

(l)

∂z

(l)

(y,u

,... , u

), l = N,P,Z

∂y

(D)

∂t

∂

∂z

∂y

(D)

∂z

(D)

(y,u

,... , u

) −

∂y

(D)

∂z

, l = D











(1)

in (−H,0) × (0, T) with additional appropriate initial

values. Here, z denotes the only remaining, verti-

cal spatial coordinate, and H the depth of the water

column. The terms Q

(l)

are the biogeochemical cou-

pling (or source-minus-sink) terms for the four tracers

and u = (u

,... , u

) is the vector of unknown physi-

cal and biological parameters. The sinking term is

only apparent in the equation for detritus. In the

one-dimensional model no advection term is used,

since a reduction to vertical advection would make no

sense. Thus, the circulation data (taken from an ocean

model) are the turbulent mixing coefﬁcient κ = κ(z,t)

and the temperature Θ = Θ(z,t), which goes into the

nonlinear coupling terms Q

(l)

but is omitted in the no-

tation.

The continuous model (1) is discretized and

solved using an operator splitting method (Marchuk,

1982), an explicit euler timestepping scheme for the

nonlinear coupling term Q and the sinking term while

using an implicit euler timestepping scheme for the

diffusion term. For further details we refer the reader

to (Prieß et al., 2011).

In the original model, the time step denoted by τ,

is chosen as one hour. The model with this particular

time step will be referred to as the high-ﬁdelity or ﬁne

one.

In the following, we will denote by y

≈ y(·,t

) the

discrete ﬁne model solution of the continous model

(1) in time step j (containing all tracers N,P,Z,D)

given as

= (y

)

i=1,...,I

, j = 1,.. . , M, y ∈ R

(2)

where I = 66 × 4 denotes the number of spatial dis-

crete points for all tracers given by 66, the number of

spatial discrete points per tracer, times 4, the num-

ber of tracers (cf. (1)) and where M = 8760 time

steps/year × 5 years denotes the number of discrete

time steps for each tracer.

2.2 Low-ﬁdelity Model

The low-ﬁdelity (or coarse) model, which is a less ac-

curate but computationally cheap representation of y

is obtained by using a coarser time discretization

given as

τ = βτ (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

the corresponding number of discrete time steps by

M = M/β, i.e., we have

= ( ˆy

)

i=1,...,I

, j = 1,... ,

y ∈ R

. (4)

Note that the parameters u for this coarse model are

the same as for the ﬁne model.

The low-ﬁdelity model is used to create the sur-

rogate of the high-ﬁdelity model, subsequently ex-

ploited to optimize the latter at a low computational

cost (cf. Section 4).

3 OPTIMIZATION PROBLEM

The key task in parameter optimization is to mini-

mize a least-squares type cost function measuring the

misﬁt between the discrete model output y = y(u)

and given observational data y

(Banks and Kunisch,

1989; Tarantola, 2005). In most cases, the problem is

IMPROVED SURROGATE-BASED OPTIMIZATION OF A MARINE ECOSYSTEM MODEL USING RESPONSE

CORRECTION

451

constrained by parameter bounds. Thus the parameter

optimization problem can be written as

min

u∈U

J( y(u)) (5)

where

J( y) :=

||y− y

:= {u ∈ R

: b

≤ u ≤ b

∈ R

, b

< b

The inequalities in the deﬁnition of the set U

of ad-

missible parameters are meant component-wise. The

functional J may additionally include a regularization

term for the parameters, which was not necessary in

our case.

Additional constraints on the state variable y

might be necessary, e.g., to ensure non-negativity of

the temperature or of the concentrations of biogeo-

chemical quantities. In our example model however,

by using appropriate parameter bounds b

and b

non-negativity of the state variables can be ensured.

This was already observed and used in (R¨uckelt et al.,

2010).

4 SURROGATE-BASED

OPTIMIZATION

For many nonlinear optimization problems, a high

computational cost of evaluating the objective func-

tion and its sensitivity, and, in some cases, the lack

of sensitivity information, is a major bottleneck. The

need for decreasing the computational cost of the op-

timization process is especially important while han-

dling complex three-dimensional models.

Surrogate-based optimization (Bandler et al.,

2004; Forrester and Keane, 2009; Leifsson and

Koziel, 2010; Queipo et al., 2005) is a methodology

that addresses these issues by replacing the original

high-ﬁdelity model y by a surrogate, in the following

denoted by s, a computationally cheap and yet reason-

ably accurate representation of y.

Surrogates can be created by approximating sam-

pled high-ﬁdelity model data (functional surrogates).

Popular techniques include polynomial regression,

kriging, artiﬁcial neural networks and support vector

regression (Queipo et al., 2005; Smola and Sch¨olkopf,

2004; Simpson et al., 2001). Another possibility,

exploited in this work, is to construct the surro-

gate model through appropriate correction/alignment

of a low-ﬁdelity or coarse model (physically-based

surrogates) (Søndergaard, 2003). The advantage of

physically-based surrogates is that a reasonable accu-

racy can be obtained using a limited number of high-

ﬁdelity model data. Also, generalization capability of

the physically-based models is typically much better

than for functional ones. The speciﬁc correction tech-

nique exploited in this work is described below.

The surrogate model is updated at each iteration

k of the optimization algorithm, typically using avail-

able high-ﬁdelity model data. The next iterate, u

k+1

is obtained by optimizing the surrogate s

, i.e.,

k+1

= argmin

u∈U

J( s

(u)). (6)

Then, the updated surrogate s

k+1

is determined by

re-aligning the low-ﬁdelity model at u

k+1

and opti-

mized again as in (6). The process of aligning the

coarse model to obtain the surrogate and subsequent

optimization of this surrogate is repeated until a user-

deﬁned termination condition is satisﬁed.

If the surrogate s

satisﬁes so-called zero-order

and ﬁrst-order consistency conditions (Conn et al.,

2000; Koziel et al., 2010) with the high-ﬁdelity

model at u

, i.e., s

) = y(u

), s

′

) = y

′

) the

surrogate-based scheme (6) is provably convergent to

at least a local optimum of (5), provided that both the

low- and high-ﬁdelity models are sufﬁciently smooth,

and the surrogate optimization step is enhanced by the

the trust-region (TR) safeguard (Conn et al., 2000;

Koziel et al., 2010). The surrogate model utilized

in this work only satisﬁes the zero-order consistency

with the high-ﬁdelity model. Still, as demonstrated in

Section 5, the performance of our surrogate-based op-

timization process is satisfactory even without using

the trust-region convergence safeguards.

4.1 Surrogate Model using Basic

Multiplicative Response Correction

The multiplicative response correction is one reason-

able way to construct a physically-based surrogate for

the marine ecosystem model given in Section 2.1.

This approach was successfully exploited in (Prieß

et al., 2011), and it is brieﬂy recalled below.

The surrogate response s

(u), at iteration k of

the optimization process, is generated by multiplica-

tive correction of the smoothed low-ﬁdelity model re-

sponse (cf. Subsection 2.2), denoted by

y, yielding

kji

(u) := A

kji

ˆy

(u),

kji

˜y

)

ˆy

)











k = 1,2,.. .

j = 1,.. . ,

i = 1,. . . ,I

β = M/

(7)

where A

kji

denotes the correction factor given as

the point wise division of the smoothed and down-

sampled ﬁne model response, denoted by

, by the

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

452

500 1000 1500 2000 2500 3000 3500 4000

0.1

0.2

0.3

time [ hours ]

DET [mmol N m

−3

]

˜y

)

ˆy(u

)

˜y

(

)

ˆy(

)

(

)

0 500 1000 1500 2000 2500 3000 3500

0.1

0.2

0.3

time [ hours ]

DET [mmol N m

−3

]

Figure 2: Surrogate’s, ﬁne (down-sampled, smoothed) and coarse (smoothed) model responses s

and

y for the tracer

detritus (at depth z ≈ −2.68m) at two points u

and corresponding perturbations

(see the text for details). The surrogate,

established at u

, is perfectly aligned with the ﬁne model at u

and provides a reasonable approximation of the ﬁne model

response at

. For illustration, only the model responses for one representative tracers, depth layer, and a part of the whole

time interval is shown.

smoothed coarse model response at the point u

. This

simple correction scheme is justiﬁed by the fact that

the overall ”shape” of the low-ﬁdelity model response

resembles that of the high-ﬁdelity one.

It was observed that smoothing allows us to re-

move the numerical noise from the coarse model re-

sponse and identify the main characteristics of the

traces of interest. Consequently, also the (down-

sampled) ﬁne model response is smoothed in (7),

yielding

, before calculating the multiplicative cor-

rection factor. Sampling of the ﬁne model response

was necessary to make it commensurable with the

corresponding coarse model response. The sampled

ﬁne model response y

is given as

:= y

βj,i

, j = 1,...,

M, i = 1,.. . , I,

∈ R

. (8)

The correction step in iteration k on the whole discrete

state vector is given as

(u) := A

◦

y(u), s

∈ R



kji



j,i

∈ R

M×I

(9)

where A

, the correction matrix in step k, and the op-

eration “◦” are deﬁned by (7).

By deﬁnition the surrogate model is zero-order

consistent with the (down-sampled and smoothed)

ﬁne model in the point u

(i.e. s

) ≈

)).

As we do not use sensitivity information, the ﬁrst-

order consistency condition cannot be satisﬁed ex-

actly. Nevertheless, as was shown in (Prieß et al.,

2011), this surrogate model exhibits quite good gen-

eralization capability, which means that the surro-

gate provides a reasonable approximation of the high-

ﬁdelity one in the neighborhood of u

Figure 2 shows the surrogate’s, ﬁne (down-

sampled) and coarse model responses

y,s

at two

different points, u

and

. The surrogate model is

established at u

and, therefore, its output is perfectly

aligned with the ﬁne model output at u

. The surro-

gate model prediction is still good at

. Since the dis-

tance between subsequent iterations points normally

decrease upon convergence of the optimization algo-

rithm, the prediction of the surrogate model is becom-

ing more and more accurate towards the end of the

optimization run.

4.2 Difﬁculties of Basic Surrogate

Formulation

Occasionally, when using the surrogate given in (7),

there might occur a situation where the coarse model

response is close to zero (and maybe even negative

due to approximation errors) and a few magnitudes

smaller than the ﬁne one, which leads to large (possi-

bly negative) entries in the corresponding correction

tensor A

. While such a correction tensor ensures

zero-order consistency at the point where it was es-

tablished (i.e., u

), it may lead to (locally) poor ap-

proximation in the vicinity of u

Figure 3 (left) illustrates these issues by showing

the smoothed surrogate’s, ﬁne (down-sampled) and

coarse model responses

y,s

for the state detri-

tus at one illustrative time interval and depth layer.

Shown are the model responses at the same iterations

and its neighborhood

∈ B

) as in Figure 2.

It should be pointed out that the overall shape

of the surrogate’s response provides a reasonable ap-

proximation of the ﬁne model response (and more ac-

curate than the corresponding coarse model response)

despite of the distortion illustrated in Figure 3. This

is supported by the fact that even without addressing

these issues, the surrogate-based scheme (6) was able

to yield satisfactory results, not only with respect to

the quality of the ﬁnal solution, but, most importantly,

in terms of the low computational cost of the opti-

mization process. This was demonstrated in (Prieß

et al., 2011).

IMPROVED SURROGATE-BASED OPTIMIZATION OF A MARINE ECOSYSTEM MODEL USING RESPONSE

CORRECTION

453

3.7 3.8 3.9 4 4.1 4.2

x 10

0.1

0.2

time [ hours ]

DET [mmol N m

−3

]

˜y

)

ˆy(u

)

˜y

(

)

ˆy(

)

(

)

3.7 3.8 3.9 4 4.1 4.2

x 10

0.1

0.2

time [ hours ]

DET [mmol N m

−3

]

Figure 3: Surrogate’s, ﬁne (down-sampled, smoothed) and coarse (smoothed) model responses s

and

y for the same

representative tracer, depth layer and parameter vectors u

and

as in Figure 2 while showing a different time interval.

Using the basic surrogate formulation (9), possible large positive and negative entries in the corresponding correction tensor

may lead to (locally) poor approximation of the resulting surrogate in the vicinity of u

(left). However, the overall shape

of the surrogate still provides a reasonable approximation of the ﬁne one (and more accurate than the corresponding coarse

model response). Employing the improvements in (10) the large positive and negative peaks are removed (right). See the text

for details.

4.3 Improved Response Correction

Scheme

The response distortion described in the previous sec-

tion is problematic towards the end of the surrogate-

based optimization run when the small changes of the

model parameters and the corresponding responses

are considered. The ”‘spikes”’ appearing in the re-

sponse due to large values of the correction term can

be viewed, in a way, as a numerical noise that slows

down the algorithm convergence and makes the opti-

mum more difﬁcult to locate.

A few simple means described below can address

these issues and further improve the accuracy of the

surrogate’s response as well as the performance of the

optimization algorithm.

We introduce non-negative bounds for the coarse

model response (the negative response is non-

physical and is a result of numerical errors due to

using large time steps in the coarse model), upper

bounds for the correction factor as well as restrict the

correction factor to one in case the ﬁne and coarse

model responses are below a certain threshold ε.

Here, we use ε = 10

−10

. More speciﬁcally, the fol-

lowing modiﬁcations of the model outputs and the

scaling factors are performed for each iteration k, dis-

crete time step j and depth layer i:

(i) ˆy

) = max{ ˆy

),1e− 8 }

(ii) A

kji

= min{ A

kji

,10 }

(iii) A

kji

= 1 if



˜y

) ≤ ε and

ˆy

) ≤ ε



(10)

where (i) is employed before smoothing the coarse

model response.

Figure 3 (right) shows the surrogate’s, ﬁne (down-

sampled) and coarse model response for the same il-

lustrative tracer, time interval and depth layer as Fig-

ure 3 (left), however, while employing the improve-

ments given in (10). It can be observed that the large

positive and negative peaks present in the surrogate

responses of Figure 3 (left) are removed after apply-

ing (10).

The numerical results presented in Section 5

demonstrate that this improved response correction

scheme allows us to further improve the computa-

tional efﬁciency of the surrogate-based scheme (6).

5 NUMERICAL RESULTS

The optimization setup used in this work is the follow-

ing. For all optimization runs we use the MATLAB

function

fmincon

, exploiting the active-set algorithm.

The following cost functions

J( z) := || z− y

∑

i=1

∑

j=1

− (y

)

, (11)

J( z) := || z−

∑

i=1

∑

j=1

− ( ˜y

)

, (12)

)

:= y

), z ∈ R

were used for the ﬁne model optimization ((11) with

z = y

), for the coarse model ((12) with z =

y) and

surrogate optimization ((12) with z = s

), whereas

(11) was used in the termination condition and to

compare the results. The down-sampled ﬁne model

output is given by (8) and the target data y

– as a ﬁrst

illustration – was synthetically created by the (down-

sampled) ﬁne model output at parameter vector u

Sampling was necessary to yield a comparable ﬁne

MATLAB is a registered trademark of The MathWorks,

Inc., http://www.mathworks.com

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

454

0 50 100 150 200 250 300 350 400

x 10

# of equivalent fine model evaluations

J ( y

(u) )

fine model opt.

coarse model opt.

SBO, original correction scheme

SBO, improved correction scheme

0 10 20 30 40 50 60

5000

10000

15000

≈96 %

≈84 %

Figure 4: Values of the cost function J (cf. (11)) versus the equivalent number of ﬁne model evaluations for a SBO run

using the surrogate model exploiting the original (cf. (7)) and the improved correction scheme (cf. (10)), as well as for a ﬁne

and coarse model optimization run. The ﬁgure also indicates those points in the SBO runs that correspond to a termination

condition of J( y

)) ≤ 50, ensuring good visual agreement between the ﬁne model output and the target. After employing

the improvements suggested in (10), the number of equivalent ﬁne model evaluations required to satisfy this termination

condition was reduced from 60 down to only 17, resulting in an increase of the corresponding time savings, from 84% to

about 96%, when compared to the direct ﬁne model optimization.

model optimization run while in (12) smoothing of

the target data is performed since smoothing of the

coarse model and surrogate’s response was employed

in the corresponding optimization runs.

For the sake of comparison, we run the direct

ﬁne and coarse model optimizations as well as the

surrogate-based algorithm (cf. (6)) exploiting the

original and improved response correction scheme (7)

and (10).

Results are presented in Figure 4 showing the

value of the cost function J (cf. (11)) versus the equiv-

alent 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 (Prieß

et al., 2011). Equivalent ﬁne model evaluations are

determined taking into account the coarsening fac-

tor β. More speciﬁcally, one evaluation of the coarse

model with a coarsening factor β is equivalent to 1/β

evaluations of the ﬁne model. The total optimization

cost is calculated as n

+ n

/β, where n

) denotes

the overall number of ﬁne (coarse) model evaluations

during the optimization run. Recall that SBO scheme

6 requires one ﬁne model evaluation per algorithm it-

eration.

Figure 4 indicates the points in the SBO run that

correspond to a termination condition of J( y

)) ≤

50. This particular value was selected as it ensures

good visual agreement between the ﬁne model output

and the target.

Figure 5 shows the down-sampled ﬁne model re-

sponse for the optimal parameter values obtained us-

ing the SBO algorithm with the original and improved

response correction scheme (denoted by u

∗

s,1

∗

s,2

Only two tracers at a certain depth level and time in-

terval are included for illustration. For the sake of

completeness the responses obtained through the di-

rect ﬁne and coarse model optimization, denoted by

∗

, are also included.

It should be noted that the model parameters ob-

tained by directly optimizing the coarse model result

in a cost function value of J(y

(

∗

)) ≈ 2960 (opti-

mization cost: 11 equivalent ﬁne model evaluations)

(cf. Figure 4). This solution is far away from that ob-

tained by the direct ﬁne model optimization (cf. Fig-

ure 5), which indicates that the coarse model is not a

reliable prediction tool.

Direct ﬁne model optimization yields a very low

cost function of J(y

(

∗

)) ≈ 1.267 · 10

−2

, corre-

sponding to a solution close to the target data (cf. Fig-

ure 5). However,the optimization cost is substantially

higher: 983 ﬁne model evaluations. Note that for bet-

ter readability, Figure 4 only shows the range 0-400

function evaluations.

In (Prieß et al., 2011), we demonstrated that in

the SBO run based on the original response correc-

tion scheme (7), the chosen termination condition

J( y

)) ≤ 50 could be reached after approximately

60 equivalent ﬁne model evaluations. This resulted

in a reduction of the total optimization cost of about

84% when compared to the ﬁne model optimization

(the direct ﬁne model optimization required 375 eval-

uations to reach this cost function value, cf. Figure 4).

After employing the improvements suggested in

IMPROVED SURROGATE-BASED OPTIMIZATION OF A MARINE ECOSYSTEM MODEL USING RESPONSE

CORRECTION

455

500 1000 1500 2000 2500 3000

0.5

1.5

time [ days ]

DIN [mmol N m

−3

]

)

∗

)

(

∗

)

∗

)

∗

)

500 1000 1500 2000 2500 3000

0.1

0.2

0.3

0.4

0.5

time [ hours ]

DET [mmol N m

−3

]

Figure 5: Fine model output y

(down-sampled) for state dissolved inorganic nitrogen (left) and the state detritus (right) at

depth z ≈ −2.68m. Shown are, in the legend from top to bottom: (i) Target y

, (ii) ﬁne model output at the initial value u

, (iii)

at the result of the direct ﬁne model optimization yielding u

∗

, (iv) at the coarse model optimum

∗

and (iv), (v) at the optima

∗

obtained by the surrogate-based algorithm (6) exploting the original (cf. (7)) and the improved (cf. (10)) response

correction scheme. Solutions (iv) and (v) are both very close to (iii) but the solution (v) was obtained at the computational

cost three times lower than (iv).

(10), only 17 equivalent ﬁne model evaluations were

required to satisfy the same termination condition,

which is over three times less than for the original

response correction scheme. The corresponding re-

duction of the total optimization cost, compared to the

direct ﬁne model optimization, is about 96% (cf. Fig-

ure 4). The corresponding solution is close to that

obtained by direct ﬁne model optimization as shown

in Figure 5.

6 CONCLUSIONS

Parameter optimization in climate models can be very

expensive in terms of the cost function and gradient

evaluations, especially for three-dimensional cases.

Therefore, methods that aim at reducing the optimiza-

tion cost of such high-ﬁdelity (ﬁne) models, such as

surrogate-based optimization (SBO) techniques, are

highly desirable. Here, the idea is to replace the ﬁne

model in the optimization run by a surrogate, a com-

putationally cheap and yet reasonably accurate repre-

sentation.

As a case study, we are interested in a parameter

optimization of a one-dimensional representative of a

class of marine ecosystem models. It follows that a

simple multiplicative response correction applied to

a temporarily coarser discretized physics-based low-

ﬁdelity (coarse) model of the system of interest is

sufﬁcient to create a reliable surrogate of the orig-

inal, high-ﬁdelity ecosystem model. This approach

allowed us to yield remarkably good results, both in

terms of the quality of the ﬁnal solution and, most im-

portantly, in terms of the relative reduction in the total

optimization cost, about 84% when compared to the

direct ﬁne model optimization.

In this paper, we demonstrated that the correction

scheme can be enhanced to alleviate the difﬁculties of

its original version, which results in further improve-

ment of the surrogate model accuracy and overall per-

formance of the optimization algorithm utilizing this

surrogate. The optimization cost was reduced by a

factor of three (from 16% to 5% of the direct high-

ﬁdelity model optimization optimization cost), which

corresponds to the cost savings of 95%.

ACKNOWLEDGEMENTS

The authors would like to thank Andreas Oschlies,

IFM Geomar, Kiel.

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