SYLLABUS Previous: 5.4 A scheme for
Up: 5 MONTE-CARLO METHOD
Next: 5.6 Computer quiz
5.5 When should you use Monte Carlo methods?
Slide : [
high dimensions -
complex BC -
parallelization ||
VIDEO
login]
Particle methods generally are efficient for problems dealing with a large
number of dimensions and complex geometries: table 5.5.0#tab.1
suggests that Monte Carlo methods become more efficient than
FD or FEM when 
(in the limit of
and neglecting computer hardware issues).
Table 5.5.0#tab.1:
Accuracy for a problem with 
-dimensions.
| Method |
Accuracy |
|---|
| Monte Carlo |
, 
is the number of particles. |
FEM/FD (
:th-order) |
,
is number of grid points. |
|
The MC method is also well suited for problems with complicated
boundary conditions.
Imagine for example a 3D cube with a bouncing ball inside. Let the cube
contain a non collisional gas, so that the atoms bounce back from the
surfaces of the cube and the ball. For simplicity let the cube and the
ball have infinite mass during the collisions with the gas particles.
The distribution of the gas atoms is fairly easy to compute with the
MC method, but nearly untraceable with a fluid method.
Parallelization is easy and efficient if the problem is linear and the
particles are decoupled: just run a copy of the simulation program on
several machines simultaneously.
The final result is obtained by summation and normalization of the results
from the individual machines, since the Monte Carlo time stepping is a
diagonal linear operation
![$\displaystyle f(t+dt)=\mathcal{L}[f(t)]=\mathcal{L}[\sum_j f_j(t)] = \sum_{ij} \mathcal{L}_{ij}[f_j(t)]=\sum_i \mathcal{L}_{ii}[f_i(t)]$](s5img95.gif) |
(1) |
where 
is either the particle or its
projection (1.4.7#eq.3).
Be careful, however, to seed the random number generators differently
on the each machines, or the simulations will all be identical!
Unfortunately, the MC method is not that easy to parallelize
for non-linear problems: if 
and 
are functions of the density
distribution, the continuous density distribution function needs to
be approximated after each step. This will dramatically reduce the
performance on a parallel machine.
SYLLABUS Previous: 5.4 A scheme for
Up: 5 MONTE-CARLO METHOD
Next: 5.6 Computer quiz
|
