<s>
Parareal	B-Operating_System
is	O
a	O
parallel	B-Operating_System
algorithm	I-Operating_System
from	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
used	O
for	O
the	O
solution	O
of	O
initial	O
value	O
problems	O
.	O
</s>
<s>
Runge-Kutta	B-Algorithm
or	O
multi-step	B-Algorithm
methods	O
,	O
some	O
of	O
the	O
computations	O
in	O
Parareal	B-Operating_System
can	O
be	O
performed	O
in	B-Operating_System
parallel	I-Operating_System
and	O
Parareal	B-Operating_System
is	O
therefore	O
one	O
example	O
of	O
a	O
parallel-in-time	O
integration	O
method	O
.	O
</s>
<s>
While	O
historically	O
most	O
efforts	O
to	O
parallelize	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
of	O
partial	O
differential	O
equations	O
focussed	O
on	O
the	O
spatial	O
discretization	O
,	O
in	O
view	O
of	O
the	O
challenges	O
from	O
exascale	B-General_Concept
computing	I-General_Concept
,	O
parallel	O
methods	O
for	O
temporal	O
discretization	O
have	O
been	O
identified	O
as	O
a	O
possible	O
way	O
to	O
increase	O
concurrency	O
in	O
numerical	B-Application
software	I-Application
.	O
</s>
<s>
Because	O
Parareal	B-Operating_System
computes	O
the	O
numerical	B-Algorithm
solution	I-Algorithm
for	O
multiple	O
time	O
steps	O
in	B-Operating_System
parallel	I-Operating_System
,	O
it	O
is	O
categorized	O
as	O
a	O
parallel	O
across	O
the	O
steps	O
method	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
approaches	O
using	O
parallelism	B-Operating_System
across	O
the	O
method	O
like	O
parallel	O
Runge-Kutta	B-Algorithm
or	O
extrapolation	O
methods	O
,	O
where	O
independent	O
stages	O
can	O
be	O
computed	O
in	B-Operating_System
parallel	I-Operating_System
or	O
parallel	O
across	O
the	O
system	O
methods	O
like	O
waveform	O
relaxation	O
.	O
</s>
<s>
Parareal	B-Operating_System
can	O
be	O
derived	O
as	O
both	O
a	O
multigrid	O
method	O
in	O
time	O
method	O
or	O
as	O
multiple	B-Algorithm
shooting	I-Algorithm
along	O
the	O
time	O
axis	O
.	O
</s>
<s>
Both	O
ideas	O
,	O
multigrid	O
in	O
time	O
as	O
well	O
as	O
adopting	O
multiple	B-Algorithm
shooting	I-Algorithm
for	O
time	O
integration	O
,	O
go	O
back	O
to	O
the	O
1980s	O
and	O
1990s	O
.	O
</s>
<s>
Parareal	B-Operating_System
is	O
a	O
widely	O
studied	O
method	O
and	O
has	O
been	O
used	O
and	O
modified	O
for	O
a	O
range	O
of	O
different	O
applications	O
.	O
</s>
<s>
The	O
objective	O
is	O
to	O
calculate	O
numerical	B-General_Concept
approximations	I-General_Concept
to	O
the	O
exact	O
solution	O
using	O
a	O
serial	O
time-stepping	O
method	O
(	O
e.g.	O
</s>
<s>
Runge-Kutta	B-Algorithm
)	O
that	O
has	O
high	O
numerical	O
accuracy	O
(	O
and	O
therefore	O
high	O
computational	O
cost	O
)	O
.	O
</s>
<s>
The	O
problem	O
with	O
this	O
(	O
and	O
the	O
reason	O
for	O
attempting	O
to	O
solve	O
in	B-Operating_System
parallel	I-Operating_System
in	O
the	O
first	O
place	O
)	O
solution	O
is	O
that	O
it	O
is	O
computationally	O
infeasible	O
to	O
calculate	O
in	O
real-time	O
.	O
</s>
<s>
Instead	O
of	O
using	O
a	O
single	O
processor	O
to	O
solve	O
the	O
initial	O
value	O
problem	O
(	O
as	O
is	O
done	O
with	O
classical	O
time-stepping	O
methods	O
)	O
,	O
Parareal	B-Operating_System
makes	O
use	O
of	O
processors	O
.	O
</s>
<s>
The	O
aim	O
to	O
is	O
to	O
use	O
processors	O
to	O
solve	O
smaller	O
initial	O
value	O
problems	O
(	O
one	O
on	O
each	O
time	O
slice	O
)	O
in	B-Operating_System
parallel	I-Operating_System
.	O
</s>
<s>
For	O
example	O
,	O
in	O
a	O
MPI	B-Application
based	O
code	O
,	O
would	O
be	O
the	O
number	O
of	O
processes	O
,	O
while	O
in	O
an	O
OpenMP	B-Application
based	O
code	O
,	O
would	O
be	O
equal	O
to	O
the	O
number	O
of	O
threads	B-Operating_System
.	O
</s>
<s>
Parareal	B-Operating_System
makes	O
use	O
of	O
a	O
second	O
time-stepping	O
method	O
to	O
solve	O
this	O
initial	O
value	O
problem	O
in	B-Operating_System
parallel	I-Operating_System
,	O
referred	O
to	O
as	O
the	O
coarse	O
solver	O
.	O
</s>
<s>
Having	O
a	O
coarse	O
solver	O
that	O
is	O
much	O
less	O
computationally	O
expensive	O
than	O
the	O
fine	O
solver	O
is	O
the	O
key	O
to	O
achieving	O
parallel	O
speed-up	O
with	O
Parareal	B-Operating_System
.	O
</s>
<s>
Henceforth	O
,	O
we	O
will	O
denote	O
the	O
Parareal	B-Operating_System
solution	O
at	O
time	O
and	O
iteration	O
by	O
.	O
</s>
<s>
Next	O
,	O
run	O
the	O
fine	O
solver	O
on	O
each	O
of	O
the	O
time	O
slices	O
,	O
in	B-Operating_System
parallel	I-Operating_System
,	O
from	O
the	O
most	O
up-to-date	O
solution	O
values	O
:	O
</s>
<s>
Now	O
update	O
the	O
parareal	B-Operating_System
solution	O
values	O
sequentially	O
using	O
the	O
predictor-corrector	O
:	O
</s>
<s>
If	O
this	O
critertion	O
is	O
not	O
satisfied	O
,	O
subsequent	O
iterations	O
can	O
then	O
be	O
run	O
by	O
applying	O
the	O
fine	O
solver	O
in	B-Operating_System
parallel	I-Operating_System
and	O
then	O
the	O
predictor-corrector	O
.	O
</s>
<s>
Note	O
that	O
other	O
stopping	O
criterion	O
do	O
exist	O
and	O
have	O
been	O
successfully	O
tested	O
in	O
Parareal	B-Operating_System
.	O
</s>
<s>
Parareal	B-Operating_System
should	O
reproduce	O
the	O
solution	O
that	O
is	O
obtained	O
by	O
the	O
serial	O
application	O
of	O
the	O
fine	O
solver	O
and	O
will	O
converge	O
in	O
a	O
maximum	O
of	O
iterations	O
.	O
</s>
<s>
For	O
Parareal	B-Operating_System
to	O
provide	O
speedup	B-Operating_System
,	O
however	O
,	O
it	O
has	O
to	O
converge	O
in	O
a	O
number	O
of	O
iterations	O
significantly	O
smaller	O
than	O
the	O
number	O
of	O
time	O
slices	O
,	O
i.e.	O
</s>
<s>
In	O
the	O
Parareal	B-Operating_System
iteration	O
,	O
the	O
computationally	O
expensive	O
evaluation	O
of	O
can	O
be	O
performed	O
in	B-Operating_System
parallel	I-Operating_System
on	O
processing	O
units	O
.	O
</s>
<s>
Typically	O
,	O
some	O
form	O
of	O
Runge-Kutta	B-Algorithm
method	I-Algorithm
is	O
chosen	O
for	O
both	O
coarse	O
and	O
fine	O
integrator	O
,	O
where	O
might	O
be	O
of	O
lower	O
order	O
and	O
use	O
a	O
larger	O
time	O
step	O
than	O
.	O
</s>
<s>
Under	O
some	O
assumptions	O
,	O
a	O
simple	O
theoretical	O
model	O
for	O
the	O
speedup	B-Operating_System
of	O
Parareal	B-Operating_System
can	O
be	O
derived	O
.	O
</s>
<s>
Although	O
in	O
applications	O
these	O
assumptions	O
can	O
be	O
too	O
restrictive	O
,	O
the	O
model	O
still	O
is	O
useful	O
to	O
illustrate	O
the	O
trade	O
offs	O
that	O
are	O
involved	O
in	O
obtaining	O
speedup	B-Operating_System
with	O
Parareal	B-Operating_System
.	O
</s>
<s>
This	O
is	O
typically	O
not	O
exactly	O
true	O
when	O
an	O
implicit	O
method	O
is	O
used	O
,	O
because	O
then	O
runtimes	O
vary	O
depending	O
on	O
the	O
number	O
of	O
iterations	O
required	O
by	O
the	O
iterative	B-Algorithm
solver	I-Algorithm
.	O
</s>
<s>
The	O
vanilla	O
version	O
of	O
Parareal	B-Operating_System
has	O
issues	O
for	O
problems	O
with	O
imaginary	O
eigenvalues	O
.	O
</s>
<s>
It	O
typically	O
only	O
converges	O
toward	O
the	O
very	O
last	O
iterations	O
,	O
that	O
is	O
as	O
approaches	O
,	O
and	O
the	O
speedup	B-Operating_System
is	O
always	O
going	O
to	O
be	O
smaller	O
than	O
one	O
.	O
</s>
<s>
So	O
either	O
the	O
number	O
of	O
iterations	O
is	O
small	O
and	O
Parareal	B-Operating_System
is	O
unstable	O
or	O
,	O
if	O
is	O
large	O
enough	O
to	O
make	O
Parareal	B-Operating_System
stable	O
,	O
no	O
speedup	B-Operating_System
is	O
possible	O
.	O
</s>
<s>
This	O
also	O
means	O
that	O
Parareal	B-Operating_System
is	O
typically	O
unstable	O
for	O
hyperbolic	O
equations	O
.	O
</s>
<s>
Even	O
though	O
the	O
formal	O
analysis	O
by	O
Gander	O
and	O
Vandewalle	O
covers	O
only	O
linear	O
problems	O
with	O
constant	O
coefficients	O
,	O
the	O
problem	O
also	O
arises	O
when	O
Parareal	B-Operating_System
is	O
applied	O
to	O
the	O
nonlinear	O
Navier	O
–	O
Stokes	O
equations	O
when	O
the	O
viscosity	B-Application
coefficient	O
becomes	O
too	O
small	O
and	O
the	O
Reynolds	O
number	O
too	O
large	O
.	O
</s>
<s>
Different	O
approaches	O
exist	O
to	O
stabilise	O
Parareal	B-Operating_System
,	O
one	O
being	O
Krylov-subspace	O
enhanced	O
Parareal	B-Operating_System
.	O
</s>
<s>
There	O
are	O
multiple	O
algorithms	O
that	O
are	O
directly	O
based	O
or	O
at	O
least	O
inspired	O
by	O
the	O
original	O
Parareal	B-Operating_System
algorithm	O
.	O
</s>
<s>
Originally	O
,	O
the	O
idea	O
was	O
formulated	O
for	O
the	O
parallel	O
implicit	O
time-integrator	O
PITA	O
,	O
a	O
method	O
closely	O
related	O
to	O
Parareal	B-Operating_System
but	O
with	O
small	O
differences	O
in	O
how	O
the	O
correction	O
is	O
done	O
.	O
</s>
<s>
is	O
defined	O
and	O
updated	O
after	O
every	O
Parareal	B-Operating_System
iteration	O
.	O
</s>
<s>
This	O
version	O
of	O
Parareal	B-Operating_System
can	O
also	O
stably	O
integrate	O
linear	O
hyperbolic	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
A	O
method	O
with	O
improved	O
parallel	O
efficiency	O
based	O
on	O
a	O
combination	O
of	O
Parareal	B-Operating_System
with	O
spectral	O
deferred	O
corrections	O
(	O
SDC	O
)	O
has	O
been	O
proposed	O
by	O
M	O
.	O
Minion	O
.	O
</s>
<s>
The	O
Parareal-SDC	O
hybrid	O
has	O
been	O
further	O
improved	O
by	O
addition	O
of	O
a	O
full	O
approximation	O
scheme	O
as	O
used	O
in	O
nonlinear	O
multigrid	O
.	O
</s>
<s>
Simulations	O
using	O
all	O
262,144	O
cores	O
on	O
the	O
IBM	O
BlueGene/P	B-Operating_System
system	O
JUGENE	O
showed	O
that	O
PFASST	O
could	O
produce	O
additional	O
speedup	B-Operating_System
beyond	O
saturation	O
of	O
the	O
spatial	O
tree	O
parallelisation	B-Operating_System
.	O
</s>
<s>
The	O
multigrid	O
reduction	O
in	O
time	O
method	O
(	O
MGRIT	O
)	O
generalises	O
the	O
interpretation	O
of	O
Parareal	B-Operating_System
as	O
a	O
multigrid-in-time	O
algorithms	O
to	O
multiple	O
levels	O
using	O
different	O
smoothers	O
.	O
</s>
<s>
It	O
is	O
a	O
more	O
general	O
approach	O
but	O
for	O
a	O
specific	O
choice	O
of	O
parameters	O
it	O
is	O
equivalent	O
to	O
Parareal	B-Operating_System
.	O
</s>
<s>
ParaExp	O
uses	O
exponential	B-Algorithm
integrators	I-Algorithm
within	O
Parareal	B-Operating_System
.	O
</s>
<s>
While	O
limited	O
to	O
linear	O
problems	O
,	O
it	O
can	O
produce	O
almost	O
optimal	O
parallel	O
speedup	B-Operating_System
.	O
</s>
