Accepted Manuscript
A novel atom search optimization for dispersion coefficient estimation in
groundwater
Weiguo Zhao, Liying Wang, Zhenxing Zhang
PII: S0167-739X(18)30657-5
DOI: https://doi.org/10.1016/j.future.2018.05.037
Reference: FUTURE 4212
To appear in: Future Generation Computer Systems
Received date : 23 March 2018
Revised date : 20 April 2018
Accepted date : 17 May 2018
Please cite this article as: W. Zhao, L. Wang, Z. Zhang, A novel atom search optimization for
dispersion coefficient estimation in groundwater, Future Generation Computer Systems (2018),
https://doi.org/10.1016/j.future.2018.05.037
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A Novel Atom Search Optimization for Dispersion Coefficient
Estimation in Groundwater
Weiguo Zhao
a,b
, Liying Wang
a,
*, Zhenxing Zhang
b
a. School of Water Conservancy and Hydropower, Hebei University of Engineering, Handan,
Hebei, 056021, China
b. Illinois State Water Survey, Prairie Research Institute, University of Illinois at
Urbana-Champaign, Champaign, IL 61820, USA
*Corresponding author.
E-mail addresses: wgzhao@illinois.edu (W. Zhao), wangliying@hebeu.edu.cn (L. Wang),
zhang538@illinois.edu (Z. Zhang)
Abstract: A new type of meta-heuristic global optimization methodology based on atom
dynamics is introduced. The proposed Atom Search Optimization (ASO) approach is a
population-based iterative heuristic global optimization algorithm for dealing with a diverse set of
optimization problems. ASO mathematically models and mimics the atomic motion model in
nature, where atoms interact with each other through interaction forces resulting form
Lennard-Jones potential and constraint forces resulting from bond-length potential, the algorithm
is simple and easy to implement. ASO is applied to a dispersion coefficient estimation problem,
the experimental results demonstrate that ASO can outperform other well-known approaches such
as Particle Swarm Optimization (PSO), Genetic Algorithm (GA) or Bacterial Foraging
Optimization (BFO) and that ASO is competitive with its competitors for parameter estimation
problems.
Keywords: Dispersion coefficient; parameter estimation; Global optimization; Particle swarm
optimization; Genetic algorithm; Atom search optimization; Bacterial foraging optimization
1. Introduction
Nature containing boundless secrets is rich and fantastic, it brings a great deal of inspiration
to people which can greatly contribute to social development. Intelligent algorithms (IAs) dealing
with difficult problems in science and engineering is an important branch of the inspiration from
nature. Since 1970s, a variety of nature-inspired optimization algorithms have been put forward
and applied to many real-world problems [1,2,3,4], which consist of two basic characteristics.
Firstly, they mimic evolving properties and living habit of biological systems. There are three
most representative algorithms. Genetic algorithm (GA) [5] is a well-known classic optimization
algorithm which can generally obtain high-quality solutions using mutation, crossover and
selection steps, and it turned out to be a good global optimization approach. Particle swarm
optimization (PSO) [6], which mimics social behaviors of bird flock. In PSO, every agent moves
around the search space to improve its solution, and their personal best positions and the globally
best position found so far are reserved by which their positions are updated locally and socially.
Ant colony optimization (ACO) [7], as another well-known optimization method, which simulates
foraging behaviors of real ant colonies. Essentially, ants communicate with each other by
pheromone trails through path formation, which assists them to find the shortest path signifying a
near-optimum solution. With their increasing popularity [8,9,10], quite a number of other similar
algorithms in the literature are developed, including evolutionary strategies (ES) [11], differential
evolution (DE) [12], evolutionary programming (EP) [13], memetic algorithm (MA) [14],
bacterial foraging optimization (BFO) [15], biogeography-based optimization (BBO) [16], cuckoo
search (CS) algorithm [17], artificial bee colony (ABC) [18], fruit fly optimization algorithm
(FOA) [19], etc.
Another basic characteristic of nature-inspired algorithms is that some of them are
enlightened from physical laws of different substances, among which, simulated annealing (SA)
[20] is one of the most well-known algorithms, being inspired from the annealing process used in
physical material in which a heated metal cools and freezes into a crystal texture with the
minimum energy. Along with DE, there are many others developed and successfully performed
including electromagnetism-like mechanism (EM) [21] based on attraction-repulsion mechanism
of electromagnetism, central force optimization (CFO) [22] and gravitational search algorithm
(GSA) [23] both inspired from Newton’s gravitational law, hysteretic optimization (HO) [24]
enlightened inspired from demagnetization process, Big Bang-Big Crunch (BB-BC) [13] inspired
from hypothesis of creation and destruction processes of the universe, wind driven optimization
(WDO) [25] based on the earth’s atmosphere motion, etc. Despite emerging many new
optimization approaches, there is no any one which can perform the best over all different types of
problems [26].
Groundwater environmental impact assessment is an important part of water environmental
impact assessment, it mainly emphasizes on the prediction and evaluation of discharge or recharge
water quality for groundwater quality impact to control or prevent the occurrence of groundwater
pollution [27,28,29]. The prediction and evaluation for groundwater is to establish necessary
mathematical models for water quality, and hydrodynamic dispersion coefficients are crucial and
essential parameters for establishing the models [30, 31 , 32 ]. So groundwater dispersion
coefficients are very important parameters which characterize the motion and spread of pollutants
in groundwater [33,34], consequently, it is vital for the accuracy of groundwater dispersion
coefficient which has a direct impact on the accuracy of mathematical simulations and the
authenticity of change prediction for groundwater quality. The numerical model is used to
simulate hydrogeological process whose result largely depends on the hydrogeological parameters
such as the permeability coefficient, the specific yield and so on, however, the field acquisition for
the parameters is usually very expensive and time-consuming, and they can not reflect the
parameters characteristics of the aquifer [35,36]. Aiming at this question, in recent years, many
researchers start to utilize different optimization methods to inversely estimate hydrogeological
coefficient with varying degrees of success [37, 38, 39]. Based on these successful applications
of the existing nature-inspired optimization techniques, in this paper, an entirely novel
optimization algorithm named atom search optimization (ASO) is proposed, which is successfully
applied to dispersion coefficient estimation of groundwater, and the experiment results illustrate
its superiority over its competitors.
The remainder of this paper is organized as follows. Section 2 describes ASO algorithm
along with the underlying physical equations of atomic motion in detail. Section 3 introduces the
model of dispersion coefficient estimation in groundwater. The application of ASO for a
hydrogeologic parameter estimation problem is given in Section 4. Section 5 presents some
conclusions and suggestions.
2. Atom Search Optimization
All substances are made of atoms which are moving all the time, and atomic motion follows
the classical mechanics [40]. According to Newton's second law, suppose that the force F
i
is an
interaction force and G
i
is a constraint force together jointly acted on the ith atom in an atom
system, then its acceleration related to its mass m
i
an be given as follows [41]
ii
i
i
FG
a
m
(1)
The Lennard-Jones potential (L-J) potential is used as the interaction force acting on the ith
atom from the jth atom in the dth dimension at t time, which can be written as
13 7
()
24 () () ()
() 2( ) ( )
() () () ()
ij
d
ij
d
ij ij ij
rt
tt t
Ft
trt rt t
r
(2)
and
'137
24 () () ()
() 2( ) ( )
() () ()
ij
ij ij
tt t
Ft
trt rt
(3)
1 1.12 1.24 1.5 2
-2
-1
0
1
2
3
4
r/
Potential energy U(r)
A
B
Repulsion
Attraction
L-J potential
F
’
Fig. 1 Atoms force curve
The force curve of atoms for molecular dynamics is shown in Fig. 1. It can be seen that atoms
keep a relative distance varying in a certain range all the time due to the repulsion or the attraction,
and the change amplitude of the repulsion relative to the equilibration distance (r=1.12σ) is much
greater than that of the attraction. The attraction is negative and the repulsion is positive, thus
atoms would not be convergent to a specific position. So, equation (3) cannot be used directly to
solve optimization problems. A revised version of this equation is developed as follows
'137
() () 2( ()) ( ())
ij ij ij
Ft t ht ht
(4)
where
()t
is the depth function to adjust the repulsion region or the attraction region, which can
be defined as
3
20
() (
1
1)
t
T
t
T
te
(5)
where a is the depth weight and T is the maximum number of iterations. The function behaviors of
'
F
with different values of
corresponding to the values of h ranging from 0.9 to 2 are
illustrated in Fig. 2. From the figure, for the different values of
, the repulsion occurs when the
values of
h are ranging from 0.9 to 1.12, the attraction occurs when h are between 1.12 and 2,
and the equilibration occurs when
1.12h
. The attraction gradually increases with the increase
of
h from the equilibration ( 1.12h ) and reaches a maximum ( 1.24h
) and then begins to
decrease. The attraction is approximately equal to zero when
h is greater than or equal to 2.
Therefore, in ASO, to improve the exploration, a lower limit of the repulsion with a smaller
function value is set to
1.1h and an upper limit of attraction with a larger function value is set
to
2.4h , so h is defined as
()
()
() ()
( )
() ()
()
()
ij
min
min
ij ij
min max
ij
ij
max
max
rt
hh
t
rt rt
ht h h
tt
rt
hh
t
(6)
0.9 1 1.11.12 1.24 1.5 2
-4
-3
-2
-1
0
1
2
3
4
5
h
F'(h)
η
=50
η
=30
η
=10
η
=5
η
=1
Repulsion
Attraction
1.1 1.12 1.24 1.16 1.18 1.2 1.22 1.2
4
-3
-2
-1
0
1
2
3
4
5
h
F'(h)
Fig. 2 Function behaviors of
'
F
with different values of
where
min
h and maxh are a lower and an upper limits of h respectively, and the length scale
()t
is defined as
2
()
() (),
()
ij
jKbest
ij
x
t
txt
Kt
(7)
and
