<s>
The	O
proper	B-Algorithm
generalized	I-Algorithm
decomposition	I-Algorithm
(	O
PGD	O
)	O
is	O
an	O
iterative	B-Algorithm
numerical	B-Algorithm
method	I-Algorithm
for	O
solving	O
boundary	O
value	O
problems	O
(	O
BVPs	O
)	O
,	O
that	O
is	O
,	O
partial	O
differential	O
equations	O
constrained	O
by	O
a	O
set	O
of	O
boundary	O
conditions	O
,	O
such	O
as	O
the	O
Poisson	O
's	O
equation	O
or	O
the	O
Laplace	O
's	O
equation	O
.	O
</s>
<s>
Unlike	O
POD	B-Algorithm
principal	O
components	O
,	O
PGD	O
modes	O
are	O
not	O
necessarily	O
orthogonal	O
to	O
each	O
other	O
.	O
</s>
<s>
By	O
selecting	O
only	O
the	O
most	O
relevant	O
PGD	O
modes	O
,	O
a	O
reduced	B-Algorithm
order	I-Algorithm
model	I-Algorithm
of	O
the	O
solution	O
is	O
obtained	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
PGD	O
is	O
considered	O
a	O
dimensionality	B-Algorithm
reduction	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
a	O
variational	B-Algorithm
formulation	I-Algorithm
of	O
the	O
problem	O
,	O
</s>
<s>
a	O
discretization	O
of	O
the	O
domain	B-Algorithm
in	O
the	O
style	O
of	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
</s>
<s>
a	O
numerical	O
greedy	B-Algorithm
algorithm	I-Algorithm
to	O
find	O
the	O
solution	O
.	O
</s>
<s>
The	O
most	O
implemented	O
variational	B-Algorithm
formulation	I-Algorithm
in	O
PGD	O
is	O
the	O
Bubnov-Galerkin	B-Algorithm
method	I-Algorithm
,	O
although	O
other	O
implementations	O
exist	O
.	O
</s>
<s>
The	O
discretization	O
of	O
the	O
domain	B-Algorithm
is	O
a	O
well	O
defined	O
set	O
of	O
procedures	O
that	O
cover	O
(	O
a	O
)	O
the	O
creation	O
of	O
finite	B-Application
element	I-Application
meshes	O
,	O
(	O
b	O
)	O
the	O
definition	O
of	O
basis	O
function	O
on	O
reference	O
elements	O
(	O
also	O
called	O
shape	O
functions	O
)	O
and	O
(	O
c	O
)	O
the	O
mapping	O
of	O
reference	O
elements	O
onto	O
the	O
elements	O
of	O
the	O
mesh	O
.	O
</s>
<s>
The	O
solution	O
is	O
sought	O
by	O
applying	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
,	O
usually	O
the	O
fixed	O
point	O
algorithm	O
,	O
to	O
the	O
weak	B-Algorithm
formulation	I-Algorithm
of	O
the	O
problem	O
.	O
</s>
<s>
The	O
number	O
of	O
computed	O
modes	O
required	O
to	O
obtain	O
an	O
approximation	O
of	O
the	O
solution	O
below	O
a	O
certain	O
error	O
threshold	O
depends	O
on	O
the	O
stop	O
criterium	O
of	O
the	O
iterative	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
PGD	O
avoids	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
as	O
solving	O
decoupled	O
problems	O
is	O
computationally	O
much	O
less	O
expensive	O
than	O
solving	O
multidimensional	O
problems	O
.	O
</s>
