<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
,	O
or	O
Monte	B-Algorithm
Carlo	I-Algorithm
experiments	I-Algorithm
,	O
are	O
a	O
broad	O
class	O
of	O
computational	O
algorithms	O
that	O
rely	O
on	O
repeated	O
random	O
sampling	O
to	O
obtain	O
numerical	O
results	O
.	O
</s>
<s>
The	O
underlying	O
concept	O
is	O
to	O
use	O
randomness	O
to	O
solve	O
problems	O
that	O
might	O
be	O
deterministic	B-General_Concept
in	O
principle	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
mainly	O
used	O
in	O
three	O
problem	O
classes	O
:	O
optimization	O
,	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
and	O
generating	O
draws	O
from	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
In	O
physics-related	O
problems	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
useful	O
for	O
simulating	O
systems	O
with	O
many	O
coupled	O
degrees	O
of	O
freedom	O
,	O
such	O
as	O
fluids	O
,	O
disordered	O
materials	O
,	O
strongly	O
coupled	O
solids	O
,	O
and	O
cellular	O
structures	O
(	O
see	O
cellular	O
Potts	O
model	O
,	O
interacting	O
particle	O
systems	O
,	O
McKean	O
–	O
Vlasov	O
processes	O
,	O
kinetic	O
models	O
of	O
gases	O
)	O
.	O
</s>
<s>
In	O
principle	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
can	O
be	O
used	O
to	O
solve	O
any	O
problem	O
having	O
a	O
probabilistic	O
interpretation	O
.	O
</s>
<s>
When	O
the	O
probability	O
distribution	O
of	O
the	O
variable	O
is	O
parameterized	O
,	O
mathematicians	O
often	O
use	O
a	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
(	O
MCMC	O
)	O
sampler	O
.	O
</s>
<s>
That	O
is	O
,	O
in	O
the	O
limit	O
,	O
the	O
samples	O
being	O
generated	O
by	O
the	O
MCMC	B-General_Concept
method	I-General_Concept
will	O
be	O
samples	O
from	O
the	O
desired	O
(	O
target	O
)	O
distribution	O
.	O
</s>
<s>
These	O
flows	O
of	O
probability	O
distributions	O
can	O
always	O
be	O
interpreted	O
as	O
the	O
distributions	O
of	O
the	O
random	O
states	O
of	O
a	O
Markov	O
process	O
whose	O
transition	O
probabilities	O
depend	O
on	O
the	O
distributions	O
of	O
the	O
current	O
random	O
states	O
(	O
see	O
McKean	O
–	O
Vlasov	O
processes	O
,	O
nonlinear	B-Algorithm
filtering	I-Algorithm
equation	I-Algorithm
)	O
.	O
</s>
<s>
When	O
the	O
size	O
of	O
the	O
system	O
tends	O
to	O
infinity	O
,	O
these	O
random	O
empirical	O
measures	O
converge	O
to	O
the	O
deterministic	B-General_Concept
distribution	O
of	O
the	O
random	O
states	O
of	O
the	O
nonlinear	O
Markov	O
chain	O
,	O
so	O
that	O
the	O
statistical	O
interaction	O
between	O
particles	O
vanishes	O
.	O
</s>
<s>
Despite	O
its	O
conceptual	O
and	O
algorithmic	O
simplicity	O
,	O
the	O
computational	O
cost	O
associated	O
with	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
can	O
be	O
staggeringly	O
high	O
.	O
</s>
<s>
Although	O
this	O
is	O
a	O
severe	O
limitation	O
in	O
very	O
complex	O
problems	O
,	O
the	O
embarrassingly	B-Operating_System
parallel	I-Operating_System
nature	O
of	O
the	O
algorithm	O
allows	O
this	O
large	O
cost	O
to	O
be	O
reduced	O
(	O
perhaps	O
to	O
a	O
feasible	O
level	O
)	O
through	O
parallel	B-Operating_System
computing	I-Operating_System
strategies	O
in	O
local	O
processors	O
,	O
clusters	O
,	O
cloud	O
computing	O
,	O
GPU	O
,	O
FPGA	O
,	O
etc	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
vary	O
,	O
but	O
tend	O
to	O
follow	O
a	O
particular	O
pattern	O
:	O
</s>
<s>
Given	O
that	O
the	O
ratio	O
of	O
their	O
areas	O
is	O
,	O
the	O
value	O
of	O
can	O
be	O
approximated	O
using	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
:	O
</s>
<s>
Uses	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
require	O
large	O
amounts	O
of	O
random	O
numbers	O
,	O
and	O
their	O
use	O
benefitted	O
greatly	O
from	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
,	O
which	O
were	O
far	O
quicker	O
to	O
use	O
than	O
the	O
tables	O
of	O
random	O
numbers	O
that	O
had	O
been	O
previously	O
used	O
for	O
statistical	O
sampling	O
.	O
</s>
<s>
Before	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
was	O
developed	O
,	O
simulations	O
tested	O
a	O
previously	O
understood	O
deterministic	B-General_Concept
problem	O
,	O
and	O
statistical	O
sampling	O
was	O
used	O
to	O
estimate	O
uncertainties	O
in	O
the	O
simulations	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
invert	O
this	O
approach	O
,	O
solving	O
deterministic	B-General_Concept
problems	O
using	O
probabilistic	O
metaheuristics	B-Algorithm
(	O
see	O
simulated	B-Algorithm
annealing	I-Algorithm
)	O
.	O
</s>
<s>
An	O
early	O
variant	O
of	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
was	O
devised	O
to	O
solve	O
the	O
Buffon	O
's	O
needle	O
problem	O
,	O
in	O
which	O
can	O
be	O
estimated	O
by	O
dropping	O
needles	O
on	O
a	O
floor	O
made	O
of	O
parallel	O
equidistant	O
strips	O
.	O
</s>
<s>
In	O
the	O
1930s	O
,	O
Enrico	O
Fermi	O
first	O
experimented	O
with	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
while	O
studying	O
neutron	O
diffusion	O
,	O
but	O
he	O
did	O
not	O
publish	O
this	O
work	O
.	O
</s>
<s>
In	O
the	O
late	O
1940s	O
,	O
Stanislaw	O
Ulam	O
invented	O
the	O
modern	O
version	O
of	O
the	O
Markov	B-General_Concept
Chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
method	I-General_Concept
while	O
he	O
was	O
working	O
on	O
nuclear	O
weapons	O
projects	O
at	O
the	O
Los	O
Alamos	O
National	O
Laboratory	O
.	O
</s>
<s>
Despite	O
having	O
most	O
of	O
the	O
necessary	O
data	O
,	O
such	O
as	O
the	O
average	O
distance	O
a	O
neutron	O
would	O
travel	O
in	O
a	O
substance	O
before	O
it	O
collided	O
with	O
an	O
atomic	O
nucleus	O
and	O
how	O
much	O
energy	O
the	O
neutron	O
was	O
likely	O
to	O
give	O
off	O
following	O
a	O
collision	O
,	O
the	O
Los	O
Alamos	O
physicists	O
were	O
unable	O
to	O
solve	O
the	O
problem	O
using	O
conventional	O
,	O
deterministic	B-General_Concept
mathematical	O
methods	O
.	O
</s>
<s>
A	O
colleague	O
of	O
von	O
Neumann	O
and	O
Ulam	O
,	O
Nicholas	O
Metropolis	O
,	O
suggested	O
using	O
the	O
name	O
Monte	O
Carlo	O
,	O
which	O
refers	O
to	O
the	O
Monte	O
Carlo	O
Casino	O
in	O
Monaco	B-Application
where	O
Ulam	O
's	O
uncle	O
would	O
borrow	O
money	O
from	O
relatives	O
to	O
gamble	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
were	O
central	O
to	O
the	O
simulations	O
required	O
for	O
the	O
Manhattan	O
Project	O
,	O
though	O
severely	O
limited	O
by	O
the	O
computational	O
tools	O
at	O
the	O
time	O
.	O
</s>
<s>
Von	O
Neumann	O
,	O
Nicholas	O
Metropolis	O
and	O
others	O
programmed	O
the	O
ENIAC	B-Device
computer	I-Device
to	O
perform	O
the	O
first	O
fully	O
automated	O
Monte	B-Algorithm
Carlo	I-Algorithm
calculations	I-Algorithm
,	O
of	O
a	O
fission	O
weapon	O
core	O
,	O
in	O
the	O
spring	O
of	O
1948	O
.	O
</s>
<s>
In	O
the	O
1950s	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
were	O
used	O
at	O
Los	O
Alamos	O
for	O
the	O
development	O
of	O
the	O
hydrogen	O
bomb	O
,	O
and	O
became	O
popularized	O
in	O
the	O
fields	O
of	O
physics	O
,	O
physical	O
chemistry	O
,	O
and	O
operations	O
research	O
.	O
</s>
<s>
The	O
Rand	O
Corporation	O
and	O
the	O
U.S.	O
Air	O
Force	O
were	O
two	O
of	O
the	O
major	O
organizations	O
responsible	O
for	O
funding	O
and	O
disseminating	O
information	O
on	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
during	O
this	O
time	O
,	O
and	O
they	O
began	O
to	O
find	O
a	O
wide	O
application	O
in	O
many	O
different	O
fields	O
.	O
</s>
<s>
The	O
theory	O
of	O
more	O
sophisticated	O
mean-field	O
type	O
particle	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
had	O
certainly	O
started	O
by	O
the	O
mid-1960s	O
,	O
with	O
the	O
work	O
of	O
Henry	O
P	O
.	O
McKean	O
Jr.	O
on	O
Markov	O
interpretations	O
of	O
a	O
class	O
of	O
nonlinear	O
parabolic	O
partial	O
differential	O
equations	O
arising	O
in	O
fluid	O
mechanics	O
.	O
</s>
<s>
We	O
also	O
quote	O
an	O
earlier	O
pioneering	O
article	O
by	O
Theodore	O
E	O
.	O
Harris	O
and	O
Herman	O
Kahn	O
,	O
published	O
in	O
1951	O
,	O
using	O
mean-field	O
genetic-type	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
for	O
estimating	O
particle	O
transmission	O
energies	O
.	O
</s>
<s>
Mean-field	O
genetic	B-Algorithm
type	O
Monte	O
Carlo	O
methodologies	O
are	O
also	O
used	O
as	O
heuristic	O
natural	O
search	O
algorithms	O
(	O
a.k.a.	O
</s>
<s>
metaheuristic	B-Algorithm
)	O
in	O
evolutionary	O
computing	O
.	O
</s>
<s>
The	O
origins	O
of	O
these	O
mean-field	O
computational	O
techniques	O
can	O
be	O
traced	O
to	O
1950	O
and	O
1954	O
with	O
the	O
work	O
of	O
Alan	O
Turing	O
on	O
genetic	B-Algorithm
type	O
mutation-selection	O
learning	O
machines	O
and	O
the	O
articles	O
by	O
Nils	O
Aall	O
Barricelli	O
at	O
the	O
Institute	O
for	O
Advanced	O
Study	O
in	O
Princeton	O
,	O
New	O
Jersey	O
.	O
</s>
<s>
Quantum	O
Monte	O
Carlo	O
,	O
and	O
more	O
specifically	O
diffusion	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
can	O
also	O
be	O
interpreted	O
as	O
a	O
mean-field	O
particle	O
Monte	O
Carlo	O
approximation	O
of	O
Feynman	O
–	O
Kac	O
path	O
integrals	O
.	O
</s>
<s>
The	O
origins	O
of	O
Quantum	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
often	O
attributed	O
to	O
Enrico	O
Fermi	O
and	O
Robert	O
Richtmyer	O
who	O
developed	O
in	O
1948	O
a	O
mean-field	O
particle	O
interpretation	O
of	O
neutron-chain	O
reactions	O
,	O
but	O
the	O
first	O
heuristic-like	O
and	O
genetic	B-Algorithm
type	O
particle	O
algorithm	O
(	O
a.k.a.	O
</s>
<s>
Resampled	O
or	O
Reconfiguration	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
)	O
for	O
estimating	O
ground	O
state	O
energies	O
of	O
quantum	O
systems	O
(	O
in	O
reduced	O
matrix	O
models	O
)	O
is	O
due	O
to	O
Jack	O
H	O
.	O
Hetherington	O
in	O
1984	O
In	O
molecular	O
chemistry	O
,	O
the	O
use	O
of	O
genetic	B-Algorithm
heuristic-like	O
particle	O
methodologies	O
(	O
a.k.a.	O
</s>
<s>
pruning	O
and	O
enrichment	O
strategies	O
)	O
can	O
be	O
traced	O
back	O
to	O
1955	O
with	O
the	O
seminal	O
work	O
of	O
Marshall	O
N	O
.	O
Rosenbluth	O
and	O
Arianna	B-Algorithm
W	I-Algorithm
.	I-Algorithm
Rosenbluth	I-Algorithm
.	O
</s>
<s>
The	O
use	O
of	O
Sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
in	O
advanced	O
signal	O
processing	O
and	O
Bayesian	O
inference	O
is	O
more	O
recent	O
.	O
</s>
<s>
It	O
was	O
in	O
1993	O
,	O
that	O
Gordon	O
et	O
al.	O
,	O
published	O
in	O
their	O
seminal	O
work	O
the	O
first	O
application	O
of	O
a	O
Monte	O
Carlo	O
resampling	B-General_Concept
algorithm	O
in	O
Bayesian	O
statistical	O
inference	O
.	O
</s>
<s>
We	O
also	O
quote	O
another	O
pioneering	O
article	O
in	O
this	O
field	O
of	O
Genshiro	O
Kitagawa	O
on	O
a	O
related	O
"	O
Monte	O
Carlo	O
filter	O
"	O
,	O
and	O
the	O
ones	O
by	O
Pierre	O
Del	O
Moral	O
and	O
Himilcon	O
Carvalho	O
,	O
Pierre	O
Del	O
Moral	O
,	O
André	O
Monin	O
and	O
Gérard	O
Salut	O
on	O
particle	B-Algorithm
filters	I-Algorithm
published	O
in	O
the	O
mid-1990s	O
.	O
</s>
<s>
Particle	B-Algorithm
filters	I-Algorithm
were	O
also	O
developed	O
in	O
signal	O
processing	O
in	O
1989	O
–	O
1992	O
by	O
P	O
.	O
Del	O
Moral	O
,	O
J	O
.	O
C	O
.	O
Noyer	O
,	O
G	O
.	O
Rigal	O
,	O
and	O
G	O
.	O
Salut	O
in	O
the	O
LAAS-CNRS	O
in	O
a	O
series	O
of	O
restricted	O
and	O
classified	O
research	O
reports	O
with	O
STCAN	O
(	O
Service	O
Technique	O
des	O
Constructions	O
et	O
Armes	O
Navales	O
)	O
,	O
the	O
IT	O
company	O
DIGILOG	O
,	O
and	O
the	O
(	O
the	O
Laboratory	O
for	O
Analysis	O
and	O
Architecture	O
of	O
Systems	O
)	O
on	O
radar/sonar	O
and	O
GPS	O
signal	O
processing	O
problems	O
.	O
</s>
<s>
These	O
Sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
methodologies	O
can	O
be	O
interpreted	O
as	O
an	O
acceptance-rejection	O
sampler	O
equipped	O
with	O
an	O
interacting	O
recycling	O
mechanism	O
.	O
</s>
<s>
From	O
1950	O
to	O
1996	O
,	O
all	O
the	O
publications	O
on	O
Sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
methodologies	O
,	O
including	O
the	O
pruning	O
and	O
resample	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
introduced	O
in	O
computational	B-Algorithm
physics	I-Algorithm
and	O
molecular	O
chemistry	O
,	O
present	O
natural	O
and	O
heuristic-like	O
algorithms	O
applied	O
to	O
different	O
situations	O
without	O
a	O
single	O
proof	O
of	O
their	O
consistency	O
,	O
nor	O
a	O
discussion	O
on	O
the	O
bias	O
of	O
the	O
estimates	O
and	O
on	O
genealogical	O
and	O
ancestral	O
tree	O
based	O
algorithms	O
.	O
</s>
<s>
For	O
example	O
,	O
Ripley	O
defines	O
most	O
probabilistic	O
modeling	O
as	O
stochastic	O
simulation	O
,	O
with	O
Monte	O
Carlo	O
being	O
reserved	O
for	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
and	O
Monte	O
Carlo	O
statistical	O
tests	O
.	O
</s>
<s>
Sawilowsky	O
distinguishes	O
between	O
a	O
simulation	O
,	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
and	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
:	O
a	O
simulation	O
is	O
a	O
fictitious	O
representation	O
of	O
reality	O
,	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
a	O
technique	O
that	O
can	O
be	O
used	O
to	O
solve	O
a	O
mathematical	O
or	O
statistical	O
problem	O
,	O
and	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
uses	O
repeated	O
sampling	O
to	O
obtain	O
the	O
statistical	O
properties	O
of	O
some	O
phenomenon	O
(	O
or	O
behavior	O
)	O
.	O
</s>
<s>
This	O
is	O
a	O
simulation	O
,	O
but	O
not	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
:	O
Pouring	O
out	O
a	O
box	O
of	O
coins	O
on	O
a	O
table	O
,	O
and	O
then	O
computing	O
the	O
ratio	O
of	O
coins	O
that	O
land	O
heads	O
versus	O
tails	O
is	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
of	O
determining	O
the	O
behavior	O
of	O
repeated	O
coin	O
tosses	O
,	O
but	O
it	O
is	O
not	O
a	O
simulation	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
:	O
Drawing	O
a	O
large	O
number	O
of	O
pseudo-random	O
uniform	O
variables	O
from	O
the	O
interval	O
 [ 0 , 1 ] 	O
at	O
one	O
time	O
,	O
or	O
once	O
at	O
many	O
different	O
times	O
,	O
and	O
assigning	O
values	O
less	O
than	O
or	O
equal	O
to	O
0.50	O
as	O
heads	O
and	O
greater	O
than	O
0.50	O
as	O
tails	O
,	O
is	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
of	O
the	O
behavior	O
of	O
repeatedly	O
tossing	O
a	O
coin	O
.	O
</s>
<s>
It	O
can	O
be	O
simulated	O
directly	O
,	O
or	O
its	O
average	O
behavior	O
can	O
be	O
described	O
by	O
stochastic	O
equations	O
that	O
can	O
themselves	O
be	O
solved	O
using	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
is	O
,	O
in	O
fact	O
,	O
random	O
experimentations	O
,	O
in	O
the	O
case	O
that	O
,	O
the	O
results	O
of	O
these	O
experiments	O
are	O
not	O
well	O
known	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
are	O
typically	O
characterized	O
by	O
many	O
unknown	O
parameters	O
,	O
many	O
of	O
which	O
are	O
difficult	O
to	O
obtain	O
experimentally	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
methods	O
do	O
not	O
always	O
require	O
truly	O
random	O
numbers	O
to	O
be	O
useful	O
(	O
although	O
,	O
for	O
some	O
applications	O
such	O
as	O
primality	B-Algorithm
testing	I-Algorithm
,	O
unpredictability	O
is	O
vital	O
)	O
.	O
</s>
<s>
Many	O
of	O
the	O
most	O
useful	O
techniques	O
use	O
deterministic	B-General_Concept
,	O
pseudorandom	B-Algorithm
sequences	I-Algorithm
,	O
making	O
it	O
easy	O
to	O
test	O
and	O
re-run	O
simulations	O
.	O
</s>
<s>
Sawilowsky	O
lists	O
the	O
characteristics	O
of	O
a	O
high-quality	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
:	O
</s>
<s>
Pseudo-random	B-Algorithm
number	I-Algorithm
sampling	I-Algorithm
algorithms	O
are	O
used	O
to	O
transform	O
uniformly	O
distributed	O
pseudo-random	O
numbers	O
into	O
numbers	O
that	O
are	O
distributed	O
according	O
to	O
a	O
given	O
probability	O
distribution	O
.	O
</s>
<s>
Low-discrepancy	O
sequences	O
are	O
often	O
used	O
instead	O
of	O
random	O
sampling	O
from	O
a	O
space	O
as	O
they	O
ensure	O
even	O
coverage	O
and	O
normally	O
have	O
a	O
faster	O
order	O
of	O
convergence	O
than	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
using	O
random	O
or	O
pseudorandom	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
Methods	O
based	O
on	O
their	O
use	O
are	O
called	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
In	O
an	O
effort	O
to	O
assess	O
the	O
impact	O
of	O
random	O
number	O
quality	O
on	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
outcomes	O
,	O
astrophysical	O
researchers	O
tested	O
cryptographically-secure	O
pseudorandom	B-Algorithm
numbers	O
generated	O
via	O
Intel	O
's	O
RDRAND	B-Language
instruction	O
set	O
,	O
as	O
compared	O
to	O
those	O
derived	O
from	O
algorithms	O
,	O
like	O
the	O
Mersenne	B-Algorithm
Twister	I-Algorithm
,	O
in	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
of	O
radio	O
flares	O
from	O
brown	O
dwarfs	O
.	O
</s>
<s>
RDRAND	B-Language
is	O
the	O
closest	O
pseudorandom	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
to	O
a	O
true	O
random	O
number	O
generator	O
.	O
</s>
<s>
No	O
statistically	O
significant	O
difference	O
was	O
found	O
between	O
models	O
generated	O
with	O
typical	O
pseudorandom	B-Algorithm
number	I-Algorithm
generators	I-Algorithm
and	O
RDRAND	B-Language
for	O
trials	O
consisting	O
of	O
the	O
generation	O
of	O
107	O
random	O
numbers	O
.	O
</s>
<s>
There	O
are	O
ways	O
of	O
using	O
probabilities	O
that	O
are	O
definitely	O
not	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
–	O
for	O
example	O
,	O
deterministic	B-General_Concept
modeling	O
using	O
single-point	O
estimates	O
.	O
</s>
<s>
By	O
contrast	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
sample	O
from	O
a	O
probability	O
distribution	O
for	O
each	O
variable	O
to	O
produce	O
hundreds	O
or	O
thousands	O
of	O
possible	O
outcomes	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
comparison	O
of	O
a	O
spreadsheet	O
cost	O
construction	O
model	O
run	O
using	O
traditional	O
"	O
what	O
if	O
"	O
scenarios	O
,	O
and	O
then	O
running	O
the	O
comparison	O
again	O
with	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
and	O
triangular	O
probability	O
distributions	O
shows	O
that	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
analysis	I-Algorithm
has	O
a	O
narrower	O
range	O
than	O
the	O
"	O
what	O
if	O
"	O
analysis	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
"	O
what	O
if	O
"	O
analysis	O
gives	O
equal	O
weight	O
to	O
all	O
scenarios	O
(	O
see	O
quantifying	O
uncertainty	O
in	O
corporate	O
finance	O
)	O
,	O
while	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
hardly	O
samples	O
in	O
the	O
very	O
low	O
probability	O
regions	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
especially	O
useful	O
for	O
simulating	O
phenomena	O
with	O
significant	O
uncertainty	O
in	O
inputs	O
and	O
systems	O
with	O
many	O
coupled	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
very	O
important	O
in	O
computational	B-Algorithm
physics	I-Algorithm
,	O
physical	O
chemistry	O
,	O
and	O
related	O
applied	O
fields	O
,	O
and	O
have	O
diverse	O
applications	O
from	O
complicated	O
quantum	O
chromodynamics	O
calculations	O
to	O
designing	O
heat	O
shields	O
and	O
aerodynamic	O
forms	O
as	O
well	O
as	O
in	O
modeling	O
radiation	O
transport	O
for	O
radiation	O
dosimetry	O
calculations	O
.	O
</s>
<s>
In	O
statistical	O
physics	O
,	O
Monte	O
Carlo	O
molecular	O
modeling	O
is	O
an	O
alternative	O
to	O
computational	O
molecular	O
dynamics	O
,	O
and	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
to	O
compute	O
statistical	O
field	O
theories	O
of	O
simple	O
particle	O
and	O
polymer	O
systems	O
.	O
</s>
<s>
Quantum	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
solve	O
the	O
many-body	B-Algorithm
problem	I-Algorithm
for	O
quantum	O
systems	O
.	O
</s>
<s>
In	O
experimental	O
particle	O
physics	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
for	O
designing	O
detectors	B-Algorithm
,	O
understanding	O
their	O
behavior	O
and	O
comparing	O
experimental	O
data	O
to	O
theory	O
.	O
</s>
<s>
In	O
astrophysics	O
,	O
they	O
are	O
used	O
in	O
such	O
diverse	O
manners	O
as	O
to	O
model	O
both	O
galaxy	B-Application
evolution	O
and	O
microwave	O
radiation	O
transmission	O
through	O
a	O
rough	O
planetary	O
surface	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
also	O
used	O
in	O
the	O
ensemble	B-Application
models	I-Application
that	O
form	O
the	O
basis	O
of	O
modern	O
weather	B-General_Concept
forecasting	I-General_Concept
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
widely	O
used	O
in	O
engineering	O
for	O
sensitivity	O
analysis	O
and	O
quantitative	O
probabilistic	O
analysis	O
in	O
process	O
design	O
.	O
</s>
<s>
In	O
microelectronics	O
engineering	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
applied	O
to	O
analyze	O
correlated	O
and	O
uncorrelated	O
variations	O
in	O
analog	O
and	O
digital	B-General_Concept
integrated	O
circuits	O
.	O
</s>
<s>
In	O
geostatistics	O
and	O
geometallurgy	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
underpin	O
the	O
design	O
of	O
mineral	O
processing	O
flowsheets	O
and	O
contribute	O
to	O
quantitative	O
risk	O
analysis	O
.	O
</s>
<s>
In	O
fluid	O
dynamics	O
,	O
in	O
particular	O
rarefied	O
gas	O
dynamics	O
,	O
where	O
the	O
Boltzmann	O
equation	O
is	O
solved	O
for	O
finite	O
Knudsen	O
number	O
fluid	O
flows	O
using	O
the	O
direct	B-Algorithm
simulation	I-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
method	O
in	O
combination	O
with	O
highly	O
efficient	O
computational	O
algorithms	O
.	O
</s>
<s>
In	O
autonomous	O
robotics	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
localization	I-Algorithm
can	O
determine	O
the	O
position	O
of	O
a	O
robot	O
.	O
</s>
<s>
It	O
is	O
often	O
applied	O
to	O
stochastic	O
filters	O
such	O
as	O
the	O
Kalman	O
filter	O
or	O
particle	B-Algorithm
filter	I-Algorithm
that	O
forms	O
the	O
heart	O
of	O
the	O
SLAM	B-Application
(	O
simultaneous	B-Application
localization	I-Application
and	I-Application
mapping	I-Application
)	O
algorithm	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
typically	O
used	O
to	O
generate	O
these	O
users	O
and	O
their	O
states	O
.	O
</s>
<s>
In	O
reliability	O
engineering	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
is	O
used	O
to	O
compute	O
system-level	O
response	O
given	O
the	O
component-level	O
response	O
.	O
</s>
<s>
In	O
signal	O
processing	O
and	O
Bayesian	O
inference	O
,	O
particle	B-Algorithm
filters	I-Algorithm
and	O
sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
techniques	I-Algorithm
are	O
a	O
class	O
of	O
mean-field	O
particle	O
methods	O
for	O
sampling	O
and	O
computing	O
the	O
posterior	O
distribution	O
of	O
a	O
signal	O
process	O
given	O
some	O
noisy	O
and	O
partial	O
observations	O
using	O
interacting	O
empirical	O
measures	O
.	O
</s>
<s>
The	O
Intergovernmental	O
Panel	O
on	O
Climate	O
Change	O
relies	O
on	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
in	O
probability	O
density	O
function	O
analysis	O
of	O
radiative	B-Algorithm
forcing	I-Algorithm
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
in	O
various	O
fields	O
of	O
computational	O
biology	O
,	O
for	O
example	O
for	O
Bayesian	B-General_Concept
inference	I-General_Concept
in	I-General_Concept
phylogeny	I-General_Concept
,	O
or	O
for	O
studying	O
biological	O
systems	O
such	O
as	O
genomes	O
,	O
proteins	O
,	O
or	O
membranes	O
.	O
</s>
<s>
Path	B-Algorithm
tracing	I-Algorithm
,	O
occasionally	O
referred	O
to	O
as	O
Monte	O
Carlo	O
ray	O
tracing	O
,	O
renders	O
a	O
3D	O
scene	O
by	O
randomly	O
tracing	O
samples	O
of	O
possible	O
light	O
paths	O
.	O
</s>
<s>
The	O
standards	O
for	O
Monte	B-Algorithm
Carlo	I-Algorithm
experiments	I-Algorithm
in	O
statistics	O
were	O
set	O
by	O
Sawilowsky	O
.	O
</s>
<s>
In	O
applied	O
statistics	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
may	O
be	O
used	O
for	O
at	O
least	O
four	O
purposes	O
:	O
</s>
<s>
To	O
provide	O
implementations	O
of	O
hypothesis	O
tests	O
that	O
are	O
more	O
efficient	O
than	O
exact	O
tests	O
such	O
as	O
permutation	B-General_Concept
tests	I-General_Concept
(	O
which	O
are	O
often	O
impossible	O
to	O
compute	O
)	O
while	O
being	O
more	O
accurate	O
than	O
critical	O
values	O
for	O
asymptotic	O
distributions	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
also	O
a	O
compromise	O
between	O
approximate	O
randomization	O
and	O
permutation	B-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
An	O
approximate	O
randomization	B-General_Concept
test	I-General_Concept
is	O
based	O
on	O
a	O
specified	O
subset	O
of	O
all	O
permutations	O
(	O
which	O
entails	O
potentially	O
enormous	O
housekeeping	O
of	O
which	O
permutations	O
have	O
been	O
considered	O
)	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
have	O
been	O
developed	O
into	O
a	O
technique	O
called	O
Monte-Carlo	B-Application
tree	I-Application
search	I-Application
that	O
is	O
useful	O
for	O
searching	O
for	O
the	O
best	O
move	O
in	O
a	O
game	O
.	O
</s>
<s>
Possible	O
moves	O
are	O
organized	O
in	O
a	O
search	B-Data_Structure
tree	I-Data_Structure
and	O
many	O
random	O
simulations	O
are	O
used	O
to	O
estimate	O
the	O
long-term	O
potential	O
of	O
each	O
move	O
.	O
</s>
<s>
The	O
Monte	B-Application
Carlo	I-Application
tree	I-Application
search	I-Application
(	O
MCTS	O
)	O
method	O
has	O
four	O
steps	O
:	O
</s>
<s>
Monte	B-Application
Carlo	I-Application
Tree	I-Application
Search	I-Application
has	O
been	O
used	O
successfully	O
to	O
play	O
games	O
such	O
as	O
Go	O
,	O
Tantrix	O
,	O
Battleship	O
,	O
Havannah	O
,	O
and	O
Arimaa	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
also	O
efficient	O
in	O
solving	O
coupled	O
integral	O
differential	O
equations	O
of	O
radiation	O
fields	O
and	O
energy	O
transport	O
,	O
and	O
thus	O
these	O
methods	O
have	O
been	O
used	O
in	O
global	B-Algorithm
illumination	I-Algorithm
computations	O
that	O
produce	O
photo-realistic	O
images	O
of	O
virtual	O
3D	O
models	O
,	O
with	O
applications	O
in	O
video	O
games	O
,	O
architecture	O
,	O
design	O
,	O
computer	O
generated	O
films	O
,	O
and	O
cinematic	O
special	O
effects	O
.	O
</s>
<s>
The	O
US	O
Coast	O
Guard	O
utilizes	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
within	O
its	O
computer	O
modeling	O
software	O
SAROPS	O
in	O
order	O
to	O
calculate	O
the	O
probable	O
locations	O
of	O
vessels	O
during	O
search	O
and	O
rescue	O
operations	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
is	O
commonly	O
used	O
to	O
evaluate	O
the	O
risk	O
and	O
uncertainty	O
that	O
would	O
affect	O
the	O
outcome	O
of	O
different	O
decision	O
options	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
allows	O
the	O
business	O
risk	O
analyst	O
to	O
incorporate	O
the	O
total	O
effects	O
of	O
uncertainty	O
in	O
variables	O
like	O
sales	O
volume	O
,	O
commodity	O
and	O
labour	O
prices	O
,	O
interest	O
and	O
exchange	O
rates	O
,	O
as	O
well	O
as	O
the	O
effect	O
of	O
distinct	O
risk	O
events	O
like	O
the	O
cancellation	O
of	O
a	O
contract	O
or	O
the	O
change	O
of	O
a	O
tax	O
law	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
in	I-Algorithm
finance	I-Algorithm
are	O
often	O
used	O
to	O
evaluate	O
investments	O
in	O
projects	O
at	O
a	O
business	O
unit	O
or	O
corporate	O
level	O
,	O
or	O
other	O
financial	O
valuations	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
also	O
used	O
in	O
option	O
pricing	O
,	O
default	O
risk	O
analysis	O
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
utilized	O
previous	O
published	O
National	O
Book	O
publication	O
data	O
and	O
book	O
's	O
price	O
according	O
to	O
book	O
genre	O
in	O
the	O
local	O
market	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
in	O
mathematics	O
to	O
solve	O
various	O
problems	O
by	O
generating	O
suitable	O
random	O
numbers	O
(	O
see	O
also	O
Random	O
number	O
generation	O
)	O
and	O
observing	O
that	O
fraction	O
of	O
the	O
numbers	O
that	O
obeys	O
some	O
property	O
or	O
properties	O
.	O
</s>
<s>
The	O
most	O
common	O
application	O
of	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
Deterministic	B-General_Concept
numerical	B-Algorithm
integration	I-Algorithm
algorithms	O
work	O
well	O
in	O
a	O
small	O
number	O
of	O
dimensions	O
,	O
but	O
encounter	O
two	O
problems	O
when	O
the	O
functions	O
have	O
many	O
variables	O
.	O
</s>
<s>
This	O
is	O
called	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
provide	O
a	O
way	O
out	O
of	O
this	O
exponential	O
increase	O
in	O
computation	O
time	O
.	O
</s>
<s>
A	O
refinement	O
of	O
this	O
method	O
,	O
known	O
as	O
importance	B-Algorithm
sampling	I-Algorithm
in	O
statistics	O
,	O
involves	O
sampling	O
the	O
points	O
randomly	O
,	O
but	O
more	O
frequently	O
where	O
the	O
integrand	O
is	O
large	O
.	O
</s>
<s>
To	O
do	O
this	O
precisely	O
one	O
would	O
have	O
to	O
already	O
know	O
the	O
integral	O
,	O
but	O
one	O
can	O
approximate	O
the	O
integral	O
by	O
an	O
integral	O
of	O
a	O
similar	O
function	O
or	O
use	O
adaptive	O
routines	O
such	O
as	O
stratified	O
sampling	O
,	O
recursive	O
stratified	O
sampling	O
,	O
adaptive	O
umbrella	O
sampling	O
or	O
the	O
VEGAS	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
A	O
similar	O
approach	O
,	O
the	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
uses	O
low-discrepancy	O
sequences	O
.	O
</s>
<s>
These	O
sequences	O
"	O
fill	O
"	O
the	O
area	O
better	O
and	O
sample	O
the	O
most	O
important	O
points	O
more	O
frequently	O
,	O
so	O
quasi-Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
can	O
often	O
converge	O
on	O
the	O
integral	O
more	O
quickly	O
.	O
</s>
<s>
Another	O
class	O
of	O
methods	O
for	O
sampling	O
points	O
in	O
a	O
volume	O
is	O
to	O
simulate	O
random	O
walks	O
over	O
it	O
(	O
Markov	B-General_Concept
chain	I-General_Concept
Monte	I-General_Concept
Carlo	I-General_Concept
)	O
.	O
</s>
<s>
Such	O
methods	O
include	O
the	O
Metropolis	B-Algorithm
–	I-Algorithm
Hastings	I-Algorithm
algorithm	I-Algorithm
,	O
Gibbs	B-Algorithm
sampling	I-Algorithm
,	O
Wang	B-Algorithm
and	I-Algorithm
Landau	I-Algorithm
algorithm	I-Algorithm
,	O
and	O
interacting	O
type	O
MCMC	O
methodologies	O
such	O
as	O
the	O
sequential	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
samplers	O
.	O
</s>
<s>
Many	O
problems	O
can	O
be	O
phrased	O
in	O
this	O
way	O
:	O
for	O
example	O
,	O
a	O
computer	B-Application
chess	I-Application
program	O
could	O
be	O
seen	O
as	O
trying	O
to	O
find	O
the	O
set	O
of	O
,	O
say	O
,	O
10	O
moves	O
that	O
produces	O
the	O
best	O
evaluation	O
function	O
at	O
the	O
end	O
.	O
</s>
<s>
In	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
the	O
goal	O
is	O
to	O
minimize	O
distance	O
traveled	O
.	O
</s>
<s>
The	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
is	O
what	O
is	O
called	O
a	O
conventional	O
optimization	O
problem	O
.	O
</s>
<s>
This	O
can	O
be	O
accomplished	O
by	O
means	O
of	O
an	O
efficient	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
even	O
in	O
cases	O
where	O
no	O
explicit	O
formula	O
for	O
the	O
a	O
priori	O
distribution	O
is	O
available	O
.	O
</s>
<s>
The	O
best-known	O
importance	B-Algorithm
sampling	I-Algorithm
method	O
,	O
the	O
Metropolis	B-Algorithm
algorithm	I-Algorithm
,	O
can	O
be	O
generalized	O
,	O
and	O
this	O
gives	O
a	O
method	O
that	O
allows	O
analysis	O
of	O
(	O
possibly	O
highly	O
nonlinear	O
)	O
inverse	O
problems	O
with	O
complex	O
a	O
priori	O
information	O
and	O
data	O
with	O
an	O
arbitrary	O
noise	O
distribution	O
.	O
</s>
<s>
Popular	O
exposition	O
of	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
Method	I-Algorithm
was	O
conducted	O
by	O
McCracken	O
.	O
</s>
