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In-Depth Information
The computational scheme of the HS algorithm comes from three musical opera-
tions: i) playing a note from the harmony memory; ii) playing a note randomly from
the entire note range; and iii) playing a note which is close to another one stored in
memory. Combination of these operations allows finding a musically pleasing har-
mony. These operations can be adapted into the engineering optimization problems as
follows: i) new generated variable values are selected from the harmony memory; ii)
new variable values are randomly selected from the possible random range; iii) new
variable values are further replaced with other values which are close to the values.
Combination of these operations allows searching a global optimum solution in an op-
timization framework. For continuous decision variables, the computational scheme
of HS can be given as follows [13]:
1.
Generate vectors (
xx
) randomly and put them into the memory.
2.
Generate a new solution vector
1
⋅⋅⋅
HMS
x
for each
′
x
′
:
x
′
from memory,
′
=
Rnd
(1,
HMS
)
-
With probability HMCR, select
xx
i
i
3.
Adjust the pitch
x
′ which is obtained from memory:
-
With probability PAR, change
x
′
as,
′
=± ×
xx w
d
(0;1)
.
i
i
-
With probability (1−PAR), do nothing.
-
With probability (1−HMCR), select
x
′
randomly.
j
j
4.
If
x
is better than the worst
x
in harmony memory, replace
x
with
x
.
5.
Repeat Steps 2 to 5 until the given termination criterion is satisfied.
The solution parameters of HS are: Harmony Memory Size (HMS), Harmony
Memory Considering Rate (HMCR), Pitch Adjusting Rate (PAR), and distance
bandwidth (
bw
).
The HS algorithm has recently been applied to various engineering optimization
problems including music composition [14], Sudoku puzzle [15], structural design
[16], water distribution network design [17], vehicle routing [18], dam scheduling
[19], groundwater modeling [11, 20], soil stability analysis [21], energy system dis-
patch [22], and transport energy demand modeling [23], etc. The detailed information
about the HS algorithm can be found at [24].
3 Parameter Structure Identification in Groundwater Modeling
This section defines the necessary solution steps for solving the parameter structure
identification problem in groundwater modeling. This is an important problem since
all the groundwater simulation models require these parameter structures while
numerically solving the governing flow and mass transport equations. Aquifers are
heterogeneous geological formations and distribution of their parameters is usually
unknown. Hence, inverse modeling approaches are usually used to identify the asso-
ciated parameter structures. Note that, inverse groundwater modeling problems often
give unstable results due to ill-posedness which is characterized by the instability and
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