<s>
In	O
mathematics	O
,	O
the	O
discrete	B-Algorithm
Poisson	I-Algorithm
equation	I-Algorithm
is	O
the	O
finite	B-Algorithm
difference	I-Algorithm
analog	O
of	O
the	O
Poisson	O
equation	O
.	O
</s>
<s>
In	O
it	O
,	O
the	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
takes	O
the	O
place	O
of	O
the	O
Laplace	O
operator	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Poisson	I-Algorithm
equation	I-Algorithm
is	O
frequently	O
used	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
as	O
a	O
stand-in	O
for	O
the	O
continuous	O
Poisson	O
equation	O
,	O
although	O
it	O
is	O
also	O
studied	O
in	O
its	O
own	O
right	O
as	O
a	O
topic	O
in	O
discrete	O
mathematics	O
.	O
</s>
<s>
The	O
preferred	O
arrangement	O
of	O
the	O
solution	O
vector	O
is	O
to	O
use	O
natural	B-Language
ordering	I-Language
which	O
,	O
prior	O
to	O
removing	O
boundary	O
elements	O
,	O
would	O
look	O
like	O
:	O
</s>
<s>
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
,	O
also	O
,	O
is	O
given	O
by	O
:	O
</s>
<s>
Among	O
the	O
methods	O
are	O
a	O
generalized	O
Thomas	B-Language
algorithm	I-Language
with	O
a	O
resulting	O
computational	O
complexity	O
of	O
,	O
cyclic	B-Algorithm
reduction	I-Algorithm
,	O
successive	B-Algorithm
overrelaxation	I-Algorithm
that	O
has	O
a	O
complexity	O
of	O
,	O
and	O
Fast	O
Fourier	O
transforms	O
which	O
is	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Poisson	I-Algorithm
's	I-Algorithm
equation	I-Algorithm
arises	O
in	O
the	O
theory	O
of	O
Markov	O
chains	O
.	O
</s>
<s>
It	O
appears	O
as	O
the	O
relative	O
value	O
function	O
for	O
the	O
dynamic	O
programming	O
equation	O
in	O
a	O
Markov	O
decision	O
process	O
,	O
and	O
as	O
the	O
control	B-Algorithm
variate	I-Algorithm
for	O
application	O
in	O
simulation	O
variance	O
reduction	O
.	O
</s>
