<s>
The	O
proper	B-Algorithm
orthogonal	I-Algorithm
decomposition	I-Algorithm
is	O
a	O
numerical	B-General_Concept
method	I-General_Concept
that	O
enables	O
a	O
reduction	O
in	O
the	O
complexity	O
of	O
computer	O
intensive	O
simulations	O
such	O
as	O
computational	O
fluid	O
dynamics	O
and	O
structural	B-Algorithm
analysis	I-Algorithm
(	O
like	O
crash	O
simulations	O
)	O
.	O
</s>
<s>
It	O
belongs	O
to	O
a	O
class	O
of	O
algorithms	O
called	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
(	O
or	O
in	O
short	O
model	O
reduction	O
)	O
.	O
</s>
<s>
The	O
main	O
use	O
of	O
POD	O
is	O
to	O
decompose	O
a	O
physical	O
field	O
(	O
like	O
pressure	O
,	O
temperature	O
in	O
fluid	O
dynamics	O
or	O
stress	O
and	O
deformation	O
in	O
structural	B-Algorithm
analysis	I-Algorithm
)	O
,	O
depending	O
on	O
the	O
different	O
variables	O
that	O
influence	O
its	O
physical	O
behaviors	O
.	O
</s>
<s>
As	O
its	O
name	O
hints	O
,	O
it	O
's	O
operating	O
an	O
Orthogonal	O
Decomposition	O
along	O
with	O
the	O
Principal	B-Application
Components	I-Application
of	O
the	O
field	O
.	O
</s>
<s>
As	O
such	O
it	O
is	O
assimilated	O
with	O
the	O
principal	B-Application
component	I-Application
analysis	I-Application
from	O
Pearson	O
in	O
the	O
field	O
of	O
statistics	O
,	O
or	O
the	O
singular	O
value	O
decomposition	O
in	O
linear	O
algebra	O
because	O
it	O
refers	O
to	O
eigenvalues	O
and	O
eigenvectors	O
of	O
a	O
physical	O
field	O
.	O
</s>
<s>
The	O
first	O
idea	O
behind	O
the	O
Proper	B-Algorithm
Orthogonal	I-Algorithm
Decomposition	I-Algorithm
(	O
POD	O
)	O
,	O
as	O
it	O
was	O
originally	O
formulated	O
in	O
the	O
domain	O
of	O
fluid	O
dynamics	O
to	O
analyze	O
turbulences	O
,	O
is	O
to	O
decompose	O
a	O
random	O
vector	O
field	O
u(x, t )	O
into	O
a	O
set	O
of	O
deterministic	O
spatial	O
functions	O
Φk(x )	O
modulated	O
by	O
random	O
time	O
coefficients	O
ak(t )	O
so	O
that	O
:	O
</s>
