<s>
This	O
article	O
deals	O
with	O
a	O
component	O
of	O
numerical	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
For	O
coarse	B-Algorithm
space	I-Algorithm
in	O
topology	O
,	O
see	O
coarse	O
structure	O
.	O
</s>
<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
an	O
auxiliary	O
system	O
of	O
equations	O
used	O
in	O
an	O
iterative	B-Algorithm
method	I-Algorithm
for	O
the	O
solution	O
of	O
a	O
given	O
larger	O
system	O
of	O
equations	O
.	O
</s>
<s>
A	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
basically	O
a	O
version	O
of	O
the	O
same	O
problem	O
at	O
a	O
lower	O
resolution	O
,	O
retaining	O
its	O
essential	O
characteristics	O
,	O
but	O
with	O
fewer	O
variables	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
to	O
propagate	O
information	O
throughout	O
the	O
whole	O
problem	O
globally	O
.	O
</s>
<s>
In	O
multigrid	O
methods	O
for	O
partial	O
differential	O
equations	O
,	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
typically	O
obtained	O
as	O
a	O
discretization	O
of	O
the	O
same	O
equation	O
on	O
a	O
coarser	O
grid	O
(	O
usually	O
,	O
in	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
)	O
or	O
by	O
a	O
Galerkin	B-Algorithm
approximation	I-Algorithm
on	O
a	O
subspace	O
,	O
called	O
a	O
coarse	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
finite	B-Application
element	I-Application
methods	I-Application
,	O
the	O
Galerkin	B-Algorithm
approximation	I-Algorithm
is	O
typically	O
used	O
,	O
with	O
the	O
coarse	B-Algorithm
space	I-Algorithm
generated	O
by	O
larger	O
elements	O
on	O
the	O
same	O
domain	B-Algorithm
.	O
</s>
<s>
Typically	O
,	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
corresponds	O
to	O
a	O
grid	O
that	O
is	O
twice	O
or	O
three	O
times	O
coarser	O
.	O
</s>
<s>
Coarse	B-Algorithm
spaces	I-Algorithm
(	O
coarse	O
model	O
,	O
surrogate	B-Algorithm
model	I-Algorithm
)	O
are	O
the	O
backbone	O
of	O
algorithms	O
and	O
methodologies	O
exploiting	O
the	O
space	B-Algorithm
mapping	I-Algorithm
concept	O
for	O
solving	O
computationally	O
intensive	O
engineering	O
modeling	O
and	O
design	O
problems	O
.	O
</s>
<s>
In	O
space	B-Algorithm
mapping	I-Algorithm
,	O
a	O
fine	O
or	O
high	O
fidelity	O
(	O
high	O
resolution	O
,	O
computationally	O
intensive	O
)	O
model	O
is	O
used	O
to	O
calibrate	O
or	O
recalibrate	O
—	O
or	O
update	O
on	O
the	O
fly	O
,	O
as	O
in	O
aggressive	O
space	B-Algorithm
mapping	I-Algorithm
—	O
a	O
suitable	O
coarse	O
model	O
.	O
</s>
<s>
An	O
updated	O
coarse	O
model	O
is	O
often	O
referred	O
to	O
as	O
surrogate	B-Algorithm
model	I-Algorithm
or	O
mapped	O
coarse	O
model	O
.	O
</s>
<s>
In	O
domain	B-Algorithm
decomposition	I-Algorithm
methods	I-Algorithm
,	O
the	O
construction	O
of	O
a	O
coarse	B-Algorithm
problem	I-Algorithm
follows	O
the	O
same	O
principles	O
as	O
in	O
multigrid	O
methods	O
,	O
but	O
the	O
coarser	O
problem	O
has	O
much	O
fewer	O
unknowns	O
,	O
generally	O
only	O
one	O
or	O
just	O
a	O
few	O
unknowns	O
per	O
subdomain	O
or	O
substructure	O
,	O
and	O
the	O
coarse	B-Algorithm
space	I-Algorithm
can	O
be	O
of	O
a	O
quite	O
different	O
type	O
that	O
the	O
original	O
finite	B-Application
element	I-Application
space	O
,	O
e.g.	O
</s>
<s>
piecewise	O
constants	O
with	O
averaging	O
in	O
balancing	B-Algorithm
domain	I-Algorithm
decomposition	I-Algorithm
or	O
built	O
from	O
energy	O
minimal	O
functions	O
in	O
BDDC	B-Algorithm
.	O
</s>
<s>
The	O
construction	O
of	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
in	O
FETI	B-Algorithm
is	O
unusual	O
in	O
that	O
it	O
is	O
not	O
obtained	O
as	O
a	O
Galerkin	B-Algorithm
approximation	I-Algorithm
of	O
the	O
original	O
problem	O
,	O
however	O
.	O
</s>
<s>
In	O
Algebraic	O
Multigrid	O
Methods	O
and	O
in	O
iterative	O
aggregation	O
methods	O
in	O
mathematical	O
economics	O
and	O
Markov	O
chains	O
,	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
is	O
generally	O
obtained	O
by	O
the	O
Galerkin	B-Algorithm
approximation	I-Algorithm
on	O
a	O
subspace	O
.	O
</s>
<s>
In	O
mathematical	O
economics	O
,	O
the	O
coarse	B-Algorithm
problem	I-Algorithm
may	O
be	O
obtained	O
by	O
the	O
aggregation	O
of	O
products	O
or	O
industries	O
into	O
a	O
coarse	O
description	O
with	O
fewer	O
variables	O
.	O
</s>
<s>
The	O
speed	O
of	O
convergence	O
of	O
multigrid	O
and	O
domain	B-Algorithm
decomposition	I-Algorithm
methods	I-Algorithm
for	O
elliptic	O
partial	O
differential	O
equations	O
without	O
a	O
coarse	B-Algorithm
problem	I-Algorithm
deteriorates	O
with	O
decreasing	O
mesh	O
step	O
(	O
or	O
decreasing	O
element	O
size	O
,	O
or	O
increasing	O
number	O
of	O
subdomains	O
or	O
substructures	O
)	O
,	O
thus	O
making	O
a	O
coarse	B-Algorithm
problem	I-Algorithm
necessary	O
for	O
a	O
scalable	B-Architecture
algorithm	O
.	O
</s>
