Biomedical Engineering Reference
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sorting time O.ln.N //. Smith et al. [
15
] compared these two scheduling methods
and found that in simulations of a polymeric system, the single-event scheduling
approach is more efficient than multievent scheduling due to avoiding the insertion
and deletion of superfluous potential collisions in the priority tree. Next, we discuss
several additional optimization approaches.
2.2
Fine Grid
In DMD, the majority of calculation is the re-evaluation of collision times between
a colliding atom and its neighbors. When the dimension of the cell (l
c
IR
max
,
the maximum interaction range) is large compared to the hardcore diameter, as
in soft sphere systems (Fig.
1
a), the number of atoms in each cell is often more
than one. As discussed above, the number of atoms in the neighboring 27 cells
is approximately 27 l
c
,where is the number density. However, assuming l
c
approximately equal to the interaction range, the number of atoms inside the
interaction range is
.4=3/l
c
whichismuchlessthan27 l
c
3
. Therefore, many
unnecessary atom pairs are included in the current scheme. We propose to divide
each cell into a finer grid with each dimension divided evenly by a number, N
f
(e.g., N
f
D
3 in Fig.
2
b). For each cell, we assign an integer address .C
x
;C
y
;C
z
/.
If the two cells have the address difference .C
x
;C
y
;C
z
/ and
X
f
C
d
1; 0
g
l
c
=N
f
/
2
<l
c
;
.max
(1)
d
D
x;y;
z
we consider the two cells as neighbors, and hence the atoms inside the cells are
neighbors. Here, l
c
d
are the cell dimensions. As N
f
increases, the number of atoms
inside the neighboring cells asymptotically approaches .4 =3/l
c
3
, approximately
16% of the original number of neighboring atoms. As the result, the computational
efficiency under the new scheme can be increased by as much as 6.4 times. On the
other hand, the frequency of cell crossing and the corresponding CPU time spent are
correspondingly increasing with this increase in N
f
. Therefore, it is possible to find
an optimal number of N
f
for each type of DMD simulation system. In our all-atom
protein model for DMD simulations, we use N
f
D
6.
We find that in dense-packing
cases such as folded proteins, we can improve the simulation efficiency by three to
four times by using a finer grid
.
2.3
Reduce the Unnecessary Square Root Calculation
The most expensive calculation in the DMD algorithm is performed after each
collision, when the DMD algorithm re-evaluates the collision times between the
colliding atoms and their neighboring atoms. Because of the costly square root
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