Information Technology Reference
In-Depth Information
M
is considered to be composed of
l
parts,
M
=
∪
l
g
=1
M
g
,
forming a layered
structure so, that each
m
g
=(
i,j,k
)
g
∈ M
g
has a single name with the same
(
i,j,k
)in each
g
-th layer.
Let the values of
u
g
(
t,x,y,z
)
,g
=1
,...,l,
be the solutions of (9), which are
taken as reference. Then the problem of constructing a CA, which simulates the
same process is stated as follows. Given a PDE in the form of (10)and an initial
array
Ω
R
(0)=
{
(
u
m
(0)
,m
)
}
, a CA should be constructed whose evolution start-
ing from the Boolean discretization
Ω
B
(0)=
{
(
v
m
(0)
,m
)
}
of
Ω
R
(0)provides at
each
t
-th iteration for any
m ∈ M
that
u
m
(
t
)
−v
m
(
t
)
<,
(11)
where
v
is the averaged value over a certain averaging neighborhood. The latter
may be different in different layers, so,
N
Av
(
m
g
)=
{
(
u
j
(
m
g
)
,φ
j
(
m
g
)) :
j
=
0
,...,q
g
}
with
φ
j
(
m
g
)
∈ M
g
.
As it was mentioned above, the transition rule of a resulting CA is a combina-
tion of two procedures: 1)computation of the next state of a standard part and
2)updating it according to the functions
f
1
,...,f
l
values. The first procedure
follows the chosen standard CA-model, which is not described here, but a repre-
sentative example is given in detail in the next section. The updating procedure
relies upon the same probabilistic rule than that used for Boolean discretization
of a real function, but with the account that the function value constitutes only
a part of the total averaged cell state.
If
f
g
(
v
m
1
(
t
)
,...,v
m
l
(
t
)) =
f
g
m
>
0, then the updating should increase the
amount of ones in
N
Av
(
m
g
). Hence, a cell (0
,m
g
)may, probably, be changed into
(1
,m
g
). Since in any averaging neighborhood
N
Av
(
m
g
)there are
q
g
(1
− w
m
g
)
zeros, then the probability of that change is
f
m
g
(
t
)
1
−w
m
g
(
t
)
.
P
(0
,m
g
)
→
(1
,m
g
)
=
(12)
When
f
m
g
<
0 the updating should decrease the resulting averaged value,
which is done by changing the cell (1
,m
g
)into (0
,m
g
)with the probability
P
(1
,m
g
)
→
(0
,m
g
)
=
|f
m
g
(
t
)
|
w
m
g
(
t
)
.
(13)
Denoting the right-hand sides of (12)and (13)as
T
+
(
f
m
g
)and
T
−
(
f
m
g
),
respectively, the updating procedure is as follows.
1
, fw
m
g
(
t
)=0
,f
m
g
(
t
)
>
0
,T
+
(
f
m
g
)
> rand
(1)
,
0
, fw
m
g
(
t
)=1
,f
m
g
(
t
)
<
0
,T
−
(
f
m
g
)
> rand
(1)
,
v
m
g
(
t
)otherwise
v
m
g
(
t
+1)=
(14)
where
rand
(1)is a random number from the interval (0,1.
Let the result of a standard CA's
t
th iteration in each
g
th layer be
Ω
S
(
t
)=
{
(
w
g
(
t
)
,m
g
)
}
and its averaged form -
Ω
S
(
t
)=
{
(
w
g
(
t
)
,m
g
)
}
, then the updating
