<s>
Model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
(	O
MOR	O
)	O
is	O
a	O
technique	O
for	O
reducing	O
the	O
computational	O
complexity	O
of	O
mathematical	O
models	O
in	O
numerical	B-Application
simulations	I-Application
.	O
</s>
<s>
Many	O
modern	O
mathematical	O
models	O
of	O
real-life	O
processes	O
pose	O
challenges	O
when	O
used	O
in	O
numerical	B-Application
simulations	I-Application
,	O
due	O
to	O
complexity	O
and	O
large	O
size	O
(	O
dimension	O
)	O
.	O
</s>
<s>
Model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
aims	O
to	O
lower	O
the	O
computational	O
complexity	O
of	O
such	O
problems	O
,	O
for	O
example	O
,	O
in	O
simulations	O
of	O
large-scale	O
dynamical	O
systems	O
and	O
control	O
systems	O
.	O
</s>
<s>
By	O
a	O
reduction	O
of	O
the	O
model	O
's	O
associated	O
state	O
space	O
dimension	O
or	O
degrees	O
of	O
freedom	O
,	O
an	O
approximation	O
to	O
the	O
original	O
model	O
is	O
computed	O
which	O
is	O
commonly	O
referred	O
to	O
as	O
a	O
reduced	B-Algorithm
order	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
Reduced	B-Algorithm
order	I-Algorithm
models	I-Algorithm
are	O
useful	O
in	O
settings	O
where	O
it	O
is	O
often	O
unfeasible	O
to	O
perform	O
numerical	B-Application
simulations	I-Application
using	O
the	O
complete	O
full	O
order	O
model	O
.	O
</s>
<s>
This	O
can	O
be	O
due	O
to	O
limitations	O
in	O
computational	O
resources	O
or	O
the	O
requirements	O
of	O
the	O
simulations	O
setting	O
,	O
for	O
instance	O
real-time	B-General_Concept
simulation	I-General_Concept
settings	O
or	O
many-query	O
settings	O
in	O
which	O
a	O
large	O
number	O
of	O
simulations	O
needs	O
to	O
be	O
performed	O
.	O
</s>
<s>
Examples	O
of	O
Real-time	B-General_Concept
simulation	I-General_Concept
settings	O
include	O
control	O
systems	O
in	O
electronics	O
and	O
visualization	B-Application
of	O
model	O
results	O
while	O
examples	O
for	O
a	O
many-query	O
setting	O
can	O
include	O
optimization	O
problems	O
and	O
design	O
exploration	O
.	O
</s>
<s>
In	O
order	O
to	O
be	O
applicable	O
to	O
real-world	O
problems	O
,	O
often	O
the	O
requirements	O
of	O
a	O
reduced	B-Algorithm
order	I-Algorithm
model	I-Algorithm
are	O
:	O
</s>
<s>
A	O
small	O
approximation	B-Algorithm
error	I-Algorithm
compared	O
to	O
the	O
full	O
order	O
model	O
.	O
</s>
<s>
constrained	O
lumping	O
of	O
polynomial	O
differential	O
equations	O
)	O
it	O
is	O
possible	O
to	O
have	O
a	O
null	O
approximation	B-Algorithm
error	I-Algorithm
,	O
resulting	O
in	O
an	O
exact	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
Contemporary	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
techniques	O
can	O
be	O
broadly	O
classified	O
into	O
5	O
classes	O
:	O
</s>
<s>
Proper	B-Algorithm
orthogonal	I-Algorithm
decomposition	I-Algorithm
methods	O
.	O
</s>
<s>
However	O
,	O
this	O
approach	O
is	O
not	O
often	O
the	O
topic	O
of	O
discussion	O
in	O
the	O
context	O
of	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
as	O
it	O
is	O
a	O
general	O
method	O
in	O
science	O
,	O
engineering	O
,	O
and	O
mathematics	O
.	O
</s>
<s>
pyMOR	O
:	O
pyMOR	O
is	O
a	O
software	O
library	O
for	O
building	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
applications	O
with	O
the	O
Python	O
programming	O
language	O
.	O
</s>
<s>
Empirical	O
gramians	O
can	O
be	O
computed	O
for	O
linear	O
and	O
nonlinear	O
control	O
systems	O
for	O
purposes	O
of	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
,	O
uncertainty	O
quantification	O
or	O
system	O
identification	O
.	O
</s>
<s>
KerMor	O
:	O
An	O
object-oriented	O
MATLAB©	O
library	O
providing	O
routines	O
for	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
of	O
nonlinear	O
dynamical	O
systems	O
.	O
</s>
<s>
MORLAB	O
:	O
Model	B-Algorithm
Order	I-Algorithm
Reduction	I-Algorithm
Laboratory	O
.	O
</s>
<s>
This	O
toolbox	O
is	O
a	O
collection	O
of	O
MATLAB/OCTAVE	O
routines	O
for	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
of	O
linear	O
dynamical	O
systems	O
based	O
on	O
the	O
solution	O
of	O
matrix	O
equations	O
.	O
</s>
<s>
libROM	O
:	O
Collection	O
of	O
C++	O
classes	O
that	O
compute	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
and	O
hyper-reduction	O
for	O
systems	O
of	O
partial	O
and	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
libROM	O
includes	O
scalable	O
and	O
parallel	O
,	O
adaptive	O
methods	O
for	O
proper	B-Algorithm
orthogonal	I-Algorithm
decomposition	I-Algorithm
,	O
parallel	O
,	O
non-adaptive	O
methods	O
for	O
hyper-reduction	O
,	O
and	O
randomized	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
Webpage	O
can	O
be	O
found	O
at	O
:	O
,	O
where	O
you	O
can	O
find	O
many	O
examples	O
,	O
e.g.	O
,	O
reduced	B-Algorithm
order	I-Algorithm
models	I-Algorithm
for	O
Lagrangian	O
hydrodynamics	O
with	O
shock-moving	O
wave	O
.	O
</s>
<s>
Model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
finds	O
application	O
within	O
all	O
fields	O
involving	O
mathematical	O
modelling	O
and	O
many	O
reviews	O
exist	O
for	O
the	O
topics	O
of	O
electronics	O
,	O
fluid	O
,	O
hydrodynamics	O
,	O
structural	O
mechanics	O
,	O
Boltzmann	O
equation	O
,	O
and	O
design	O
optimization	O
.	O
</s>
<s>
The	O
first	O
usage	O
of	O
model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
techniques	O
dates	O
back	O
to	O
the	O
work	O
of	O
Lumley	O
in	O
1967	O
,	O
where	O
it	O
was	O
used	O
to	O
gain	O
insight	O
into	O
the	O
mechanisms	O
and	O
intensity	O
of	O
turbulence	O
and	O
large	O
coherent	O
structures	O
present	O
in	O
fluid	O
flow	O
problems	O
.	O
</s>
<s>
Model	B-Algorithm
order	I-Algorithm
reduction	I-Algorithm
also	O
finds	O
modern	O
applications	O
in	O
aeronautics	O
to	O
model	O
the	O
flow	O
over	O
the	O
body	O
of	O
aircraft	O
.	O
</s>
